Bars in Lincoln close at 1am.
You can buy booze in Lincoln at any grocery store 24/7, except on Sunday mornings until noon. Not sure about Gretna/Omaha. Main grocery stores around here are Hy-Vee, Super Saver (in Lincoln) and Wal-Mart superstores, but I recommend avoiding the Wal-Marts. For obvious reasons.
The Haymarket is a region of downtown Lincoln from O Street to R street, 9th street to 7th street. It contains kitschy little shops and is the best concentration of restaurants in town, including The Oven, Lazlo's, Vincenzo's, El Portrero, Maggie's, Bread & Cup, Brewsky's, Buzzard Billy's, Old Chicago (which I'm sure you're familiar with), and for dessert Ivanna Cone, the best ice cream in Nebraska (7th & P, right across from Lazlo's). There are several bars down there as well, but the "main" bars are from 9th street east along O street.
You will be thoroughly unimpressed with a runza. Try one just to try it, but you won't buy two. It's a traditional old food in Nebraska that is very meh.
Valentino's pizza is slightly better than frozen pizza. 30 years ago it was fantastic, but they went corporate, lost their soul and now they make trash. Do not get their pizza. It is an absolute crime what they have turned that business into.
There are no good pizza joints in Lincoln. If you want fantastic pizza come to my house on Friday and I'll make you a pie, regular, thin-crust or deep-dish.
Misty's in the Havelock district (on Havelock Ave) is a traditional old restaurant in Husker folklore. They serve a mean prime rib and the Havelock location has a pep band every Friday night before the game. As many have mentioned, it is in dire need of renovation. The food is meat-and-potatoes, and decent. They have a downtown location at 11th & P, but I have heard way too many people talk about bad experiences there. The one time I went there I had a very bad experience with the server and I won't go back.
For tailgating I highly recommend hitting the 'gates under the viaduct west of the stadium. I go over there before every game and it's a great time.
Elk Creek Water is the shizzle. Just don't drink too much of it. The comparison to a screwdriver someone made up above is apt, but it's got about 10x the alcohol of a regular screwdriver. It tastes like a nice mix of orange and grapefruit juice and it will knock you on your keister.
I-80 between Gretna and Lincoln gets VERY congested before games, so if you're coming down I recommend coming early in the AM to avoid traffic.