Where in Ohio are you? I'm in Columbus.
Getting into Camp Perry is no problem. You just drive up to the gate, tell the MP that you're there to see the matches, and drive down towards the firing line. Be prepared to walk; parking is in short supply during the matches.
As a spectator, you're allowed to get within about 100 yards of the firing line. A good set of binoculars will allow you to see what's actually going on.
The National Matches are divided into four parts: Muzzleloading, Pistol, Smallbore (.22 rifle) and High Power (centerfire rifle). Which set of matches interests you most?