How to play this endgame without consulting the tablebases

In positions where the side with the Rook is trying to hold back the passed pawns it is often critical how quickly the King can scramble back to base, because the Rook will usually be easy. If for example the King must reach g8, then ..Kxa2 leaves the King with one rank less to cross and therefore (other things being equal) will save a tempo. Therefore my instinct would tend toward ..Kxa2 rather than ..Rxa2, even before attempting any analysis. The question whether other things truly are equal is the reason why endings in general need a lot of calculation. In this case they probably are.

However, why did White take the pawn before it promoted? 1.f5 waiting for 1..a1=Q 2. Rxa1 Kxa1. White would then be two tempi up compared to the game. He made Black spend a move on promoting, and also ensured that the Black King would have to cross all eight ranks.