I did not use Uber at the Hyatt, but I did take 3 Uber rides in San Juan over the Xmas holidays, including a pick up and drop off at the airport. I had no difficulties with any of these rides, and actually thought my drivers were friendlier and more courteous than is typical in the mainland. Obviously, 3 rides isn't a meaningful sample, but I would happily use Uber again in Puerto Rico.

If you do get picked up by Uber at SJU, pay attention to the pick up information. Uber has only been allowed to pick up at the airport for a few months, and there is still some confusion. There are 2 pick up areas. Make sure you go the right one. Signage isn't great, and if you ask someone for directions, they may send you in the wrong direction. I speak from experience on that.