I recently finished this game, which ended in a draw, but I'd like to know if it was possible to win it. I had an idea of bringing my king over to the passed pawn, bringing a rook up to trade/force white's rook back, and advancing that way but it turned out not to be possible. I'm sure the position was won about the middle game, but I let the advantage slip.

I think it is (nearly) impossible to win such a rook endgame. It was a draw already at move 45. No chance to support the advance of the a-pawn.

Maybe Vlad Kramnik can win this but none of us mortal ...

I think you're right. I had the idea of using my king to aid the cause, but when it came to it, it just didn't work.

I think white has a more comfortable position but it's drawn because black needs to block the g pawn as well