A basic rook ending

I am interested in this position:



It seems like black can't go after the a pawn with the king, but that black is winning anyway because the kingside pawns are fast and can be supported by the king. Is this true? How universal is this kind of situation where one player has an outside passed pawn that is useless because the opponent's rook is behind it?

There are all sorts of tactics that prevent black from going after the a pawn, but black wins anyway? Just has to be very careful to not allow the a8 rook to move with check. Can't walk over along the seventh rank either, due to Rf8 and skewering the black rook after it captures the pawn.

I believe it is a win by advancing the f-pawn: 36...Kg6 and 37 ...f5.
White has advanced his pawn too far. Better to stop at a6, so that there is a hole where the king can hide.

Push the f-pawn as tpr pointed out, leave the g pawn as is, so when white plays Rg8, Rxa7 will both capture the pawn and protect the g pawn.
P.S. it's a very common endgame when one side has a pawn on a7 and rook on a8.