A tricky one, but I think White can survive with 1.e4. Black can't play f1...4 because 2.e5+ actually wins for white. So after 1....fxe4 2.Kf2 I think White has enough time to pick of the e4 pawn and then attack Blacks two remaining pawns. Or I could be wrong!
Instructive Endgames

i would think doing kg3 then pushing h4 would be a good idea, just have to make sure the black h pawn doesnt have time to promote. draw at the least
The first move is absolutely essential. The remainder is simple because both sides lack several tempi to win. After white 1.e4! black must capture it as 1. ... f4? 2. e5+! wins for white - as the OP said in his analysis! If white starts with 1.Kf2? Kc6 black can ignore the sacrifice 2.e4 and respond with 2. .. f4!

Mistake. After pieces are exchanged white exchange pawns and will make a passed h pawn by playing g4, g5. Black king is not in time to prevent the passer.
White to play, Seems like Black will catch that queenside pawn and win using his kingside pawn majority. Can White survive?