Unfortunately I won't be going. As an out-of-towner, the only game I make it back for every year is homecoming. I will list a few of the favorite local "don't miss" eateries here. Your hotel desk or any local can point you in the right direction. People are very friendly in Las Cruces so you won't have any problems.
Nopalito's (5 minutes from Aggie stadium, so it's a great stop before the game). Their specialty is New Mexico style enchiladas. One of my local favorites.
Chope's (a bit off the beaten path - 20 minutes or so from NMSU. Little hole in the wall type place in the middle of nowhere, but worth the effort). Outstanding chile rellenos.
Roberto's - fast food style, but don't go there if you're really in a hurry (or for that matter any other restaurant in town. Things move in slow motion in LC as compared to California). The tacos and stuffed sopaipillas are great there.
Si Senor - a little more upscale sit-down restaurant.
Nellie's Cafe - another local mom and pop joint with outstanding Mexican food.
Dick's Cafe - great Mexican food as well as burgers and other classics.
La Posta - Not the best Mexican food in town, and locals tend to not eat there because there's better Mexican food available in town for less money. However, the chile rellenos are very good, and it's in kind of a cool settiing on an old historic plaza. It's in an old adobe stagecoach station where Billy the Kid was gunned down and General McArthur stayed a few times as a child. I would recommend it for a visitor looking for something a little out of the ordinary.
Keep in mind that Las Cruces is not at all a late night town, so there won't be much open after the game. For something local that is open late, try Farley's - a cheesy sports bar type hangout, but they do have some pretty good local eats on the menu. There's also a Hooters in town now and a few other national chains, but don't expect a thriving nightlife after the game.