Park on Campus at the garage on the corner of 25th and Highland. 2 years ago it was only like $5 and it is next to the stadium. Should be able to tailgate there. Vandy is pretty lenient with the tailgating rules.
Vanderbilt is stuck between a busy road and a residential neighborhood. Traffic is going to be bad leaving. Go get something to eat/drink on West End and come back in a few hours.
There is a large vacant spot next to the parking garage that usually has quite a few tailgaters. You can park in the garage and tailgate there. No problem.
be careful parking across the street in Centennial Park. While it might look like "open parking" the park police sometimes get pizzy and have cars towed.
There's shopping and food on West End. If you want to make a day of it you could come in the morning. I don't think there is a too early. They probably won't let you park overnight in the garage but otherwise you should be okay.