The Blackwell is on the complete other side of campus, on Woodruff Ave. (this would be "20th Ave."), and the med center is spread out over 9th-12th Ave. But there's a free CABS bus run by OSU that you can hop on without any sort of student ID (catch the Campus Loop North) with a stop across the street from the Blackwell.
P.S. The Blackwell Inn is probably pricey, Erin Andrews of ESPN stays there for any football/basketball coverage during the year. Cool fact: She is currently trying to sue Ohio State, since one of the alleged "peeps" happened at the Inn.
Any other hotels will be in downtown Columbus, which are literally down the street from campus but not really near the med center. Public buses run from downtown to campus (don't know about fare, but the #2 will be the one you need), or you could grab a taxi.
Hope that helps, good luck!
Edit: I forgot a handful of cheaper hotels (compared to downtown) on Olentangy River Rd. These are a very short taxi ride away from the med center, and probably financially more reasonable.