I also immediately thought of Panozzo's - they would be able to offer precisely what you want, and can do everything from simple meat/cheese plates all the way up to full entrees.
That being said, I'm not sure I would really consider 13th/Michigan to be "near" 35th/State, though I suppose it depends on your perspective. Something in Bridgeport would probably be more convenient.
And, while Panozzo's doesn't really break the bank, I wouldn't consider it terribly affordable, compared to places like Mariano's. The food is terrific, really high quality, and everyone there knows what they are doing - but you definitely pay for it.
If it fits your budget and the distance is manageable, I think it would be a slam dunk. I routinely grab Panozzo's for Ravinia, and everyone that comes with me raves about it.