The only place I've ever eaten late at night in downtown Indy is 
The Slippery Noodle Inn, 327 S Meridian St.  Their website says they serve 
until 2 AM on Saturdays.  Menu runs the whole gamut from nachos to filet mignon.  
But - oy vey, what great music they have there.  It's like Stevie Ray's, with a restaurant in it.
I haven't eaten there for several years; I'm not vouching for the food quality, but the menu looks sufficient for 2 AM.