interesting endgame.
Often with endgames in this scenario it either turns out a win for the side with a pawn on the h-file or else a draw. Best is to get that pawn as far up as you can! Then when black brings his king over to attack the rook blocking the pawn white will have to do the same, then if an exchange of rooks ooccur black is closer to whites pawns so he gobbles them up and then advances his pawns to win.
So black needs to first get an exchange of rooks first if possible. White should avoid the exchange.
Im pretty sure black should win here with perfect play, but mayhap not.
How should black have gone about winning this position? Note that white can still castle kingside.