Does One Piece Movie: The Desert Princess and the Pirates: Adventures in Alabasta have anything to do with the Alabasta arc in the anime series?
Yes, it is a retelling of the Alabasta arc.
From wikipedia it is "adapting a story arc from the original manga" which is of course the same arc adapted for the anime.
There are some One Piece movies or specials like this one that can be useful for rewatching or catching up on the story faster than watching the anime or reading the manga.
The other main movie "Episode of Chopper Plus: Bloom in the Winter, Miracle Sakura" is a retelling of the amazing Drum Island arc with Franky, Nico, and Sunny added in.
Then there are Episode of Sabo, Episode of East Blue, Episode of Alabasta, Episode of Merry, etc. which are television specials showing these arcs in short time frames.