Mopar Nats 2011

Zanesville is about the same distance from the show as Brice Road (maybe a few minutes farther), but its east instead of west of the track. You Don't want to be in the down town part of Zanesville, but the west end wouldn't be too bad for driving to the show. But theres not a whole lot there.
Like the others have said, the Heath and Hebron area is the place stay for the cruising, and theres plenty of restaurants there. But those hotels are likely already booked solid.
You may just end up on Brice or one of the hotels by the Columbus airport. If you stay near the airport, a couple of the hotels there have easy access to a back way out of the airport which is vital as traffic on the main road past those hotels to the airport can get real bad. For many years we stayed at one of the hotels by the airport and they always had a hose and towels available for washing cars and it was cheaper than staying on Brice.
Unless you need to go back to your room before going to the cruising, just plan on going straight from the track to Heath, and back to the room afterwards. Whether you stay on Brice Road in Reynoldsburg, or at one of the hotels by the Columbus airport, or on the west end of Zanesville, you'll be about 30 to 40 minutes drive to the show.