I've recently played Canterbury and Canton Brookside. Both are fantastic golf courses. This summer, Canterbury has had the best fws I've ever seen (no divots and the tightest turf anywhere). Brookside has the best green complexes in NE Ohio and are truly world-class. But, both are private. If you are a member of a private club, a phone call might work. But, it's not probable.
On the public side, The Quarry in Canton is a lot of fun. Fowler's Mill (Pete Dye) on the east side of cleveland is another option as well.