Here's how I think they do:
George Washington is top tier but he's ganged up near the beginning and thrown out, possibly right away, since all the presidents will know who he is.
John Adams is bottom tier. He's ignored until the end. Remains competitive and collaborates with pushing some people out, but he's easily thrown out when he is singled out.
Thomas Jefferson bottom tier. Not a fighter despite his height. He probably leaves the match when he realizes he hasn't a chance.
James Madison bottom tier. He probably leaves the match with Jefferson.
James Monroe is middle tier, but his age is catching up with him. He runs out of stamina midway and is thrown out.
John Quincy Adams is bottom tier. He realizes this is madness, so he is the first to walk out of the ring, but he stays as a spectator to write about the whole thing in his diary.
Andrew Jackson is top tier, but he's so frail that he seems unassuming. He exaggerates his age for much of the early match. Jackson is one of the main presidents, along with Jefferson and Wilson, encouraging president to gang up on the stronger presidents. Jackson aids little as he conserves his strength, while others get worn out. Towards the middle of the match, Jackson gets really scrappy and is nearly thrown out several times, but uses all sorts of cheap shots to stay in the match. He ultimately wins the entire royal rumble through survival instinct.
Martin Van Buren is bottom tier. He is ignored until the middle of the match.
William Henry Harrison is bottom tier. He's ill the entire time and reduced to aggressively coughing in people's faces. He spends the first part of the match in the middle of the mat on his back, having a coughing fit. FDR rolls over him in his wheel chair several times. He lasts, surprisingly, to the latter half of the match where he is rediscovered and casually thrown out.
John Tyler is bottom tier. Thrown out in the middle by Obama.
James K. Polk is middle tier. Not particularly strong, but he has some of that same instinct as Jackson. Originally, he is instructed by Jackson, but once Jackson views him as a future threat, he tosses Polk out after instructed Polk to throw WH Harrison out.
Zachary Taylor is top tier. Taylor is one of those ganged up at the beginning and thrown out.
Millard Fillmore is middle tier. He is treated as a bottom tier and is ignored for much of the match. He does surprisingly well, but is thrown out in the later phase of the match.
Franklin Pierce is middle tier. He is treated as top tier, but his chronic depression and alcoholism has really damaged any potential that he had. He actually falls out of the ring when he tries to climb the turnbuckle.
James Buchanan is bottom tier. He helps in throwing out the top tier presidents, but he's runs out of the ring once every middle and lower tier wrestler starts turning on each other.
Abraham Lincoln is top tier. He is ganged up on and thrown out immediately after Washington.
Andrew Johnson is middle tier. He, like Pierce, is drunk during the match. He's ignored for much of the match, but is thrown out by Jackson after Johnson refuses to listen to any more of Jackson's instructions.
Ulysses S. Grant is middle tier. While a great general, he isn't much of a brawler. Nevertheless, he last until the early part of the later phase of the match.
Rutherford B. Hayes is middle tier. He does surprisingly well is one of the last 10 presidents in the match.
James A. Garfield is middle tier. While having relative youth on his side, Garfield is so cautious as to miss many excellent opportunities. He nearly throws out Jackson, but Jackson hangs on to the ropes, and manages to twist himself in such a way that he pulls Garfield out of the ring.
Chester A. Arthur is bottom tier. He's suffering from what is probably Crohn's Disease. He spends the early part of the match showboating his mutton chops, but doubles over in pain when he's hit for the first time. He spends much of the early match laying on the ground out of fear of being hit if he stands up. He eventually just slides out of the ring and leaves.
Grover Cleveland is top tier. Cleveland has size and determination. He is one of the last 10 presidents in the ring. He's also the last president to be ganged up on.
Benjamin Harrison is bottom tier. Short, old, and not a fighter. He spends much of the match defending his grandfather, including knocking FDR over in his wheelchair when Roosevelt is rolling over WH Harrison. Cleveland throws the younger Harrison out in the middle of the match.
William McKinley is middle tier. He is thrown out towards the end of the middle phase of the match.
Theodore Roosevelt is top tier. He is ganged up on at the beginning and thrown out after Washington and Lincoln.
William Howard Taft is middle tier. His weight is his only advantage. He is conflicted because he fears hitting anyone that is looking at him as they might hit him back, and he won't hit anyone that isn't looking because he doesn't think it's honorable. During a conflicted moment, he is tossed out by Cleveland, Jackson, Polk, and Wilson. The latter has a stroke in the process of doing this.
Woodrow Wilson is bottom tier. Wilson has some of the same survival instinct of Jackson, but it's more cerebral than tenacious. Wilson, along with Jefferson and Jackson, are central in planning the strategy to toss out the best fighters. Wilson spends much of the middle of the match hyperaware of where the other fighters are. He does little physical exertion except to help others push other fighters out. He has a stroke helping throw out Taft. Jackson takes advantage of this to throw Wilson out immediately after Taft is thrown out.
Warren G. Harding is middle tier. He's not in the best of health and so his stamina runs out in the middle of the match. He's thrown out almost at his own suggestion.
Calvin Coolidge is bottom tier. Most of the wrestlers forget he's in the ring. He leaves the ring at some point but no one notices.
Herbert Hoover is middle tier. Hoover is able to defend himself until the middle of the match, but is then thrown out.
FDR does little but help push people out since he's confined to a wheel chair. He spends the early part of the match rolling over WH Harrison, Arthur, and anyone else that is on the ground. Benjamin Harrison knocks him out of the wheelchair, and FDR spends much of the middle phase trying to get back into the wheelchair. Once he's in, Trump tries to life FDR with the wheelchair but hurts his back doing so. It never occurs to Trump to take FDR out of the wheelchair first, something that Wilson does. However, Wilson hasn't the strength to throw FDR out, so he leaves that for someone else. Clinton later kicks the wheelchair out of the ring so FDR can't get back in. FDR somehow manages to the final 10 competitors, as he predominantly crawls around on the ground, but does a good job at taking out people knees. He's eventually kicked out of the ring.
Harry S Truman is middle tier. He does a fairly admirable job, but is thrown out in the middle phase.
Dwight D. Eisenhower is middle tier. He could have been top tier, but his health isn't the best. He is a surprisingly great fighter in the early phase, but by the middle he is worn out and is thrown out.
JFK is bottom tier. He looks like he's top tier, but he's in such bad health that he is almost useless. Jackson kicks him in the spine when he isn't looking and Kennedy is unable to stand up straight for the rest of the match. He's thrown out by Nixon in the late early phase of the match, soon after JFK is run over by FDR.
LBJ is top tier. LBJ is thrown out 4th as the president gang up on him. LBJ has that survival instinct, so he gets away from the gang of presidents a few times, but he eventually is caught and thrown out.
Richard Nixon is middle tier. Like Jackson, he's survivor. He spends much of the match convincing some presidents to just leave the match before they get hurt. He and Jackson are the last two presidents remaining. The duel lasts almost as long as the rest of the match lasted, and seems to be half dialogue. Nixon tries to rely on psychological attacks and defenses rather than brawling. It doesn't work.
Gerald Ford is middle tier. While formerly athletic, Ford's sense of balance is his undoing. He escapes being ganged up on, and falls out of the ring trying to climb a turnbuckle.
Jimmy Carter is bottom tier. Carter is thrown out in the early middle phase when trying to help some of the injured presidents.
Ronald Reagan is bottom tier. He would have been higher in his youth. He appears lost in the ring several times, but this seems to have been for show. Nixon figures him out and throws him out, with the help of Bill Clinton, towards the end of the middle phase.
George HW Bush is middle tier. He fights rather well, partially because he and his son work together. He is eventually thrown out in the middle phase after GW Bush had already been thrown out.
Bill Clinton is middle tier. Clinton uses psychological tricks, such as lying, in order to aid his situations. "Look! Washington's back in the ring with a chair. Look out!" and then Clinton punches his gullible foe. Clinton last deep into the wrestling match.
George W. Bush is top tier. He's much more athletic than the other presidents. Since most of the older presidents have no clue who he is, he's able to escape being ganged up on in the beginning. By the early middle rounds, it's clear that Bush is a threat to everyone, especially since he's collaborating with his father. However, Clinton tricks Bush Jr into thinking that Bush Sr is under attacks, and is able to throw Bush Jr out as he's distracted.
Barack Obama is top tier. Obama is arguably the most agile of the wrestlers. He's thrown out, however, by a collection of the pre-FDR presidents --slave holders and white supremacists--who get upset at Obama for not doing what they've been telling him to do. Woodrow Wilson and John Tyler are the worst of the lot and both are nearly disfigured by Obama. He is able to throw Tyler out before the wave of racists throw out Obama.
Donald Trump is bottom tier. Old, fat, and not particularly smart, Trump does have a survival instinct, but he lacks courage to use this instinct for brawling. He never lifts a finger unless a group of presidents are nearly throwing someone out of the ring. Nevertheless, Trump claims victory anytime someone is thrown out. He announces loudly that "he's winning" several times in the match. He is mostly ignored. When the ring is down to 15 presidents, they start paying attention to him. He accuses Obama of cheating when Obama becomes the first presidents to throw a punch at him, even though the punch doesn't land. After another missed punch is thrown by Clinton, Trump tumbles out of the ring, farts loudly upon impact with the floor, gets up, and runs out of the ring.
Forgive me for typos. Typed this fast.