I went to a game in Trashville last season and had an absolute blast! It's always fun to be in a new city and cheer on the team. From what I noticed, a lot of visiting fans, especially UM folks, seemed to stay at the Hilton Nashville Downtown. It's right in the heart of everything, so it's convenient for sightseeing and grabbing a bite with fellow fans before the game.
As for hotel blocks, UM sometimes sets up a hotel block for season ticket holders. I'd check with the ticket office or alumni association to see if they've set something up for this season.