Here you have to be very carefull, the minutest slip means a draw here.
First we have to redefine the goals of both sides.
The goal for white is pretty obvious:
The pawn must promote and the white king must keep the road to promotion clear.
White needs at least 7 moves for that.
Black must conquer the pawn. The most logical place is to intercept the pawn is at g8, which is allready under attack by the rook. This means that the rook is already standing well, this means that the black king must head for h7 or f7. h7 can be reached in 8 moves while f7 can be reached in 5 moves.
The following moves the main goals continue to be the same for both sides, but tactically every position needs a redefinition of the method to go further. White can make the road of black towards blacks goals longer, but he needs to invest tempi himself in order to do so. As long as the extra tempi on both sides don't tip the balance, black continues to be fine. In diagram 1 both 1. ... Kd3 and 1. ... Kc3 (surprisingly!) brings black closer to his goal.
You see that play becomes critical when both sides need about the same amount of tempi to reach their goals. When that amount is far off balance, play is simple and straightforward.