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I lived in Japan for a year, and did a little hiking while I was there. Only reason I didn't do more is because I didn't own a car, and I didn't find other hiking buddies until a few months into my stay. Pretty much everybody who goes camping there stay in campgrounds, which are horrifyingly crowded. To be honest, I never did figure out the legality of wild camping. Most sources said it's illegal, but some people claimed that "camping" is legal on any public land, including river banks in many cities, so who knows. If I were you, I'd do it far from the trails, as stealthily as you know how.
A car will help significantly with reaching your destination. You can arrange a combination of taxis, buses, and trains to get to many trailheads, but you'll be stuck to the bus schedule to get home at night, which may not align with your needs. The problem is that since you may not be able to get a car if you'll only be there for a short time, you can't just ask any old acquaintance you meet to drive you to go hiking. Very few Japanese people like to be in the outdoors, and Japanese hikers are a rare breed.
Expect to see a lot of people wearing the latest, most top of the line hiking gear, who have absolutely no idea what they are doing. Also, the trails can get really rough, sometimes going straight up, and sometimes etching a gigantic ditch into the hill, since "trail building" is not really a concept over there. I saw very few trails with proper switch-backs.