I think you'd have a blast. The OSU/Michigan game is on Saturday, Nov. 24th in Columbus. OSU Campus should be hopping on Friday night and Saturday. Kind of an amazing atmosphere if you go before kickoff and experience the tailgating. I'm sure some OSU grad or current student can give tips on places to go or do, but this seems to cover some of the OSU bar scene:
5 best campus bars for the not-so-college students
Beers inside Arena are about same as you would expect in any Arena. Around $13 for a 24oz of Labatts/Budweiser product or for a 16oz cup of Craft Beer.
Not much BBQ around arena district, but lots of other food/beer/pub/brewery type options. The R Bar is where most out of town fans go for a nice local hockey bar. Here's a list of Arena District hotspots:
Columbus Ohio Restaurants | Arena District
For BBQ, one option would be to take the free "CBUS" circular. It just goes up and down High Street between the Short North (which is north of Arena District) down to German Village, which is South of Arena District. Lots of places in between. The BBQ would be "Smoked on High" which is in German Village, you'd take CBUS south as far as it goes and then you walk about another block south on high.
Another option would be to go to Land Grant Brewing on Town Street. They have a Ray Ray's BBQ food truck onsite. It could be walked by taking CBUS down to Rich Street and walking West about 1/2 mile, but probably suggest Uber. The main Ray Ray's restaurant is on High Street north of OSU. Basically, you can find a lot of different restaurants, bars, etc. along High Street between German Village up to North of OSU campus. Another popular haunt by out of towners is Thurmans, known for their huge burgers. Some fun downtown spots include Pins Mechanical (Duck pin bowling/pinball/sports bar/craft beer/whiskey bar), 16-bit Barcade bar (free 80's video games with craft beers and cocktails), elevator brewing tap house, wolf's ridge brewery.
CBUS (free downtown bus) ROUTE MAP:
https://www.cota.com/wp-content/uploads/2016/04/CBUS-hotel-map.pdf
Smoked on High:
Home
Land Grant Brewery:
The Land-Grant Brewing Company
Ray Ray's BBQ:
Hog Pit
R Bar (arena district):
[R Bar] R Bar | The Best Place to Watch Hockey in Columbus.
Wolf's Ridge (go to the taproom in back if you go):
Home
Pins Mechanical:
COLUMBUS
16 Bit BArcade:
COLUMBUS
Thurmans Café:
The Thurman Cafe | Tastiest Burgers in Columbus, Ohio
Short North Nightlife:
Bars & Nightlife | Short North, Columbus Ohio