Traffic near the stadium is murder on game day and finding parking anywhere close to campus is nearly impossible (not like at Ryan Field where a 15-20 minute walk down central can yield positive results). Depending on where you are staying, my recommendation would be to hang out in the short north (plenty of cool places) and then hop on the 2 bus until you get close to Lane Ave. From there you can walk to the Shoe in about 15 minutes. There's probably a local shuttle option as well.
As Jules pointed out, campus area is kinda ******, but if you're into the rowdier college stuff then the aforementioned Varsity Club would be the place (I personally hate that establishment). There is massive re-development going down on High St. (both on and off campus) and several of the older dives across from OSU (like Bernie's) are being torn down so VC would likely be your best bet, but you could choose close tweener options like BW3's (it's an Ohio based after all) or Eddie George's as well.
Columbus is a great place to spend a weekend...a real hidden gem in the Midwest. I'd be happy to offer more thorough suggestions (there's plenty to do) related to food, drink, and entertainment for those interested in making the trek to central Ohio.