I imagine most of the larger downtown hotels have a shuttle that would run that late. 11p is a pretty popular arrival time (all the final flights from hubs everywhere arriving), so there's lots of traffic to bring back!

It's not open 24 hours, but there is a sushi place downtown called Shogun (300 South and Main) that stays open until 1a on Thursday nights and 2a on Friday and Saturday nights. The food is mediocre but the place is really popular with the late night clubbing crowd.

The kitchen at The Bayou (Cajun food, about 700 South and State) stays open until 12:30a on weekends and has the best selection of beer you will find anywhere in the state. The Bayou is actually one of my favorite places to take out-of-towners. Good food and a great atmosphere.

Both Red Rock (200 South and 200 West) and Squatters (300 South and 200 West) are open until 1a. I prefer the food at Red Rock personally.

Sadly the best 24 hour place (Bayleaf) just closed a month or two ago.