I lived in Jax for seven years. The one thing I miss as far as food goes, is the seafood. Clark's Fish Camp is good. I actually just went there for the first time last week. I never lived near it, so I tried it on a recommendation while visiting. The little 'hole-in-the-wall' takeout places are also good.
Neptune Beach is the closest beach to downtown if you want to stay on the beach. From the Arena downtown, you take the Mathews Bridge to Arlington Expressway and continue on to Atlantic Blvd., following it to Neptune Beach.
If you want to stay closer to the games, there are several places to stay off of I-95, Exit 344, Butler Blvd.
You could also stay North of downtown, near the airport.
The best shopping and dining is the Town Center, located near the intersection of Butler Blvd. and I-295. Butler Blvd. also eventually reaches Jacksonville Beach.
Jacksonville Landing has some downtown shopping and dining. It's where Tennessee had a pep rally the day before last year's bowl game in Jax.
I can recommend pizza and bbq places too, but I suggest eating some seafood while visiting the coast.