Yes, it seems to be drawn. Unfortunately the PGN comments window isn't yet active in the site's PGN viewer, but here goes. First consider the following variation, but in particular the position after
[FEN "8/8/8/3k3p/3P3K/2P3R1/8/7r w - - 0 1"]
1.Rh3 Rg1 2.Kxh5 Rg8 3.Rf3 Kc4 4.Kh4 Kd5? 5.Rf5+!
Here we don't even need to consult an engine because there is a better oracle to be had. This 6-piece position is covered by existing endgame tablebases, and you can confirm with a query that this is drawn. So we know for a fact that going this
1.Rh3 route leads to a draw. (White could try delaying the pawn capture and making a more productive immediate move, but it doesn't help, and just leads to play like that coming after
1.Kg5, discussed at the end below.)
As to the idea behind Black's play from here, it is to keep the pawns at bay with the king, and as you intimated in your original post, to use the rook to cutoff the white king as much as possible. It's difficult to say much more about the exact drawing method here, because the play can be very complicated (one reason to love endgames, by the way). To give an idea, consider the position after
4.Kh4. 10 of Black's possible moves keep the draw (including those that simply move the rook along the g-file), but the perhaps seemingly harmless
4...Kd5?, putting the king back where it was, is suddenly a loss for Black in 41 moves, but only if White plays
5.Rf5+!, as all other fifth moves again draw. One way to get a good feel for the correct and incorrect play (and for the general notion of how impenetrable optimal play in technical endgames can be) is to toy around with moves at the link I gave above.
White's only other first move option is
1.Kg5, and it's not too surprising that this draws if winning the h-pawn immediately already draws. It's true that White avoids getting his king cut off on the h-file, but the cost of leaving Black with a distracting outside passed pawn is too high. The point is that the black king and rook can easily stop the pawns if the white king isn't helping out his rook. But now if the white king does try to get involved, say via ideas like
Kf6-e6 and pushing the d-pawn, Black can keep things level because she has her own passed pawn to play with.