The Alamo will take about five minutes.
Bohanan's does a decent steak, one of the only good restaurants in downtown.
Give the the Granary a shot for BBQ. If the weather isn't too hot, you can walk to the Granary from downtown along the Riverwalk. Once the Riverwalk leaves downtown it becomes greenspace rather than commercial, pleasant walk. The walk from downtown to the Pearl (where the Granary is located) is maybe 20 minutes. Otherwise it's a five minute drive.
Definitely stop by Brown Coffee Co off I-10 on your way to UTSA. In fact, grab some breakfast tacos from the taqueria next door. 3 tacos for $1.50, get the chorizo. Then eat them at Brown with a pourover of whatever's on the menu that day.
https://goo.gl/maps/bC09w
Or grab something to go from the other location just outside downtown, across the street from the Granary. Best espresso you'll ever have.