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'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.