If you're defining "best" as most authentic to the University, I agree that City Café in Northport (dive meat & three, lunch), Rama Jama, DePalma's or one of the barbecue places would be the places to go.
As for the barbecue, I'm not getting into the debate as to which is better. I will say that the original Dreamland location in Jerusalem Heights south of town is more authentic to the University -- Bryant and other coaches used to be regulars. There's another location just across the Hugh Thomas bridge in Northport, but it's newer and its atmosphere is a bit more generic.
Both Archibald's and the original Dreamland are in obscure locations, so you might want to google directions ahead of time.
If you're defining "best" as the finest dining experience, Evangeline's, River, Side By Side and R. Davidson's Chop House would be the spots.
Stadium isn't generally open at all. You might be able to arrange a tour through the Athletic Department, but I'd be surprised -- the west side renovation is in full swing, it's mostly fenced off, and there's all sorts of construction activity, equipment, etc.
Rama Jama is right across the street from the stadium, and is open through the construction, so you might be able to kill two birds with one stone there -- eat good burgers at an authentic student place, and take a stroll around the stadium such as it is right now. Scenery from nearby sorority row isn't bad either.