The following process can be used:
1) Push the h-pawn and put your king on the h-file. The g-pawn is used as a shield from the black rook's horizontal checks.
2) Put your rook on the queenside, on either the rank the g-pawn is on right now, or on the rank above.
3) Push your g-pawn one, followed by advancing your king up one square. The picture on the kingisde looks something like Kh3, pawn on g3, and pawn on h4.
4) If Black's king is directly opposing you (say on the h5-square), check it with your rook. If Black's rook is preventing this (e.g., by being on the a5-square), then nothing is stopping you from advancing your g-pawn with g4+.
5) Then, advance the h4-pawn to h5. Now your king has a place to hide on h4 (since your g-pawn is either ready to blockade on g4, or it's about to advance to g4).
Repeat this general process. Essentially, before advancing a pawn, make sure your king can either move up and get new shelter, or that your rook is already on the same rank to prevent the king from being checked in the first place.