
Trains: As a commenter pointed out, you can take the Amtrak Crescent Line from ATL to New Orleans. It's about an 11 hour ride, but it's fairly scenic. I had a friend take the trip last year, and she said that you can bring coolers and booze on the train. She enjoyed it. The fare is $116 / person round trip, and they have 10-15% discounts for senior citizens.
As a point of comparison, you could expect to pay $150-210 per car for gas if you were to drive to New Orleans from Atlanta (total round trip) depending on the vehicle's fuel efficiency.
Automobiles: You can get rental cars in New Orleans for dirt cheap on HotWire.com. That'll get you to the game in Baton Rouge. I picked up a full-sized car for $25/day via that site. There were deals for closer to $17/day a few weeks ago for smaller vehicles.
Good luck.
PWD
(Note: none of these sites are advertisers. I wish they were though)