Mediterranean Restaurant - that's its name, not just the description - seems to be one of the oldest places downtown still thriving. Asheville is more about the shock of the new, with lots and lots of new, good, independent and very fun restaurants downtown. I can't think of any others that are really old enough to meet your criteria, especially since they demolished the old Hot Dog King, but I stand ready to be corrected.
Wherever you end up, the eating is just terrific. Between Chai Pani, Creperie Bouchon, Tupelo Honey Cafe, Lexington Ave. Brewery and Early Girl Eatery, you'll be spoiled for choice.