If your air bnb is literally downtown, I wouldn't drive to the stadium. Leave the car downtown and walk to campus across the World's Fair Park area. On a normal day the drive is usually about 5-10 minutes from downtown to Neyland Stadium. On game day it will be way longer than walking and you'll still most likely park a long way from the gates.
As for places to eat right downtown there is an amazing amount of choice and variety. JC Holdway (farm to table), Chesapeakes (seafood), Nama Sushi, Cru Bistro (American), Calhoun's on the River, Emilia's (fresh Italian), Sweet P's (bar bq), Stock & Barrel ( up scale burgers), Cafe 4, Tomato Head, Not Watsons and Tupelo Honey are all very good and in the middle of downtown with the exception of Calhoun's which is just down the river from the stadium. A few of the favorite downtown bars are Clancy's Tavern and Whiskey House (Irish), The Kerns Library (in the Oliver Hotel), Preservation Pub, Downtown Grill and Brewery, and the Bistro at the Bijou. Some other favorite Knoxville restaurants worth considering out of downtown are Naples Italian, Bistro by the Tracks, Copper Cellar, Smoky Mountain Grill and Brewery, Ye Olde Steakhouse, Littons and Connors. Of course there are lots of other places including a number of local brewers.
I hope you have fun!