There isn't much right by the stadium in terms of actual bars, the closest spot is next to the arena across the street from the baseball field on J Philip Randolph Blvd. There are numerous bars on Bay St. Downtown, a good long walk away from the stadium. There will be lots of places to quench your thirst at the field itself, though. Watching the teams warm up before the game may be of interest. The stadium will open about two hours before game time if things haven't changed too much.
There will be several kiosks and places for beer/food consumption just west of the stadium, and I would recommend you check out the whole tailgate scene going on all around Everbank Field as a true look at America. There are massive parking lots full of folks in lawn chairs around their cars, with grills and other things setup and running, kids tossing the pigskin around, frisbee throwing, and vehicles used only for game day that are customized in a particularly American way to celebrate the love for the local squad. The locals aren't very good, but if they get going and the Duuuuuuvalllll roar starts, you may see something special.
Also, you must get there in time to see the mascot, Jaxson DeVille (I'm not making this up) find another death defying way to get in the stadium. "Risking his life for your entertainment." Since it's the opener, he'll have something new up his sleeve. He's really the best thing they've had going for the last few years. He'll probably be standing on one of the light towers and wind up jumping off to float down to the field using some rigging. Curtis (his real name) is pretty awesome.
One more thing, if you've never seen the anthem done complete with a flyover, it's something you'll not soon forget. Be early enough to check it out, don't be stuck in the beer queue underneath when it happens. Great big flag, fireworks on the south end, etc. A jet fighter will buzz the stadium at low level, suck all the air out the place, and pop your eardrums in the process exactly at the time the anthem ends. It's the tits.
Enjoy. It's 'Merica, lid.