Yeah, I would advise staying in New Orleans Thursday and, at least the day on Friday. MUCH more to do and see than in Baton Rouge.

You could drive up to BR on Friday afternoon/early evening and have dinner/hangout, stay in BR that night if you'd like. But you'd probably have more fun in New Orleans and I would imagine most UCLA fans coming to the game will stay there as well. It's an easy drive from New Orleans to BR. I wouldn't take a bus unless it was a shuttle bus provided by the hotel or some other group specifically running the bus for visiting fans traveling to the game. I'm not aware of any public transportation from New Orleans to BR, and if there is, I wouldn't want to ride it. Probably not the most reputable crowd.

Depending on kickoff time (likely between 5:00 - 7:00 CT), I would get to campus at least 4-5 hours before kickoff. There will be tailgate parities everywhere, and most folks are pretty hospitable to visiting fans (particularly non-SEC fans). There will be some who are obnoxious and will ribb you a bit for being an opposing fan, but they will likely be more than happy to offer you all the food and drink you want. Many of those with large tailgate parties take great pride in their cooking skills and hospitableness, so they are proud to share/party with visitors. Just follow the rules of not being an a-hole or grabbing someone's wife's/girlfriend's arse, unsolicited, and you should be fine.

If you don't want to eat at a tailgate party, there are multiple restaurants on/near campus. The Chimes restaurant is THE LSU traditional restaurant. Great place to grab beer and cajun/creole dishes and Po-boys. Prepare to wait because it is popular with locals and visitors alike. There are other restaurants near The Chimes, including the very first Raising Cane's chicken fingers if that interests you at all. Also connected to The Chimes is the Varsity Theatre. There is generally some sort of concert/event the Friday night before the game and (probably) after the game.

On the other side of campus (walking distance from the stadium) is Walk-On's restaurant. Much "newer" than the Chimes, but it has become an LSU staple. The menu selection, in general, is about the same as The Chimes. However, it's a bit bigger and generally they will have a big outdoor tent party both the night before and night of the game. It will stay packed pretty much from open to close on game day.

About 2 hours or so before kickoff the LSU team will arrive on campus and walk down "the hill" (it isn't much of a hill) into the stadium followed by the marching band. Fans will line both sides of the street to cheer them in.

I would try to get to my seat about 20-30 minutes before kickoff. If you have ever watched LSU play you probably know the pre-game entrance of the band is noteworthy. They march out on to the field, blast the first four notes of "Hold that Tiger!", and the crowd goes wild. That's the first, and a necessary step of truly experiencing the Tiger Stadium atmosphere.