Can black win from this position?
Rook pawns never win when the defending king is in front of them. He can push you back to h1 but he cannot force you out of the corner. Eventually the advance of the pawn will stalemate you.

Hey!
Black cannot win. The white king just goes to h1 and moves to g1/h2 and back to h1. It's going to be a stalemate, if black advances the pawn to h2.
The only possibility to win this type of positions (with an a- or h-pawn) is if the black king reaches g2/b2 and the white king is somewhere on e2/d2, because the a-/h-pawn can then run under the cover of his king to become a queen.
By the way if you were wondering it is from one of my own games on this website. The end result was a draw, as he managed to take the pawn.