Haha, your trip to the US is the exact same days as my trip to London, funny
The following tips are decidedly more Minneapolis-biased, since as Minnesota mentioned, those were my stomping grounds for about 6 years. I barely know St. Paul, haha.
There's plenty to do in St. Paul, but depending on how long you'll be here, you may want to venture over to Minneapolis!
Dinkytown is alright, if you're looking for cheap beer and cheap food. College bar stuff. If that's your scene, head over to Burrito Loco in Dinkytown. Plenty of beer/drink specials, and good tacos/burritos. Mesa Pizza is another good one if you're looking for a quick/cheap bite to eat. Pizza by the slice, they have tons of quirky varieties. My personal favorites are the tortellini pizza and gyro pizza.
If you're up for waking up to watch soccer on Saturday or Sunday, I'd recommend heading over to one of the many pubs/bars in Minneapolis that show games. Brit's is the mainstay, it is a Liverpool bar. The Local is also cool, that's an Arsenal bar. The Nomad is a Spurs bar. All of them are fun, the people are always friendly and up for some banter!
I know you're a beer guy Lapa, so here's a list off the top of my personal favorite breweries that MSP has to offer, and that I'd suggest visiting:
Surly Brewing: Huge place, but usually very busy. Probably Minnesota's trendiest beer right now. Located right by the U campus.
Insight Brewing: Large taproom, great selection of beers, cool vibe. NE Minneapolis
Indeed Brewing: Smaller place, but the beer is excellent. Some of the best beer in MN right now, in my opinion. Their Rum King is one of my favorites of all time. NE Minneapolis
Dangerous Man Brewing: In an old bank building, pretty small, but bursting with character. Great beer here as well. NE Minneapolis
Lake Monster Brewing: This is a newer one, but I really dig it. Large place, never have trouble finding a spot. The beers are good, and the brewers are cool guys. Pretty much right in between Minneapolis and St. Paul
Bad Weather Brewing: One of my few St. Paul spots, this one is not far at all from the X. Really cool place, highly recommended.
There are sooooo many more, but that's what I came up with off the top of my head.