I lived in Japan for about a year. There are some truly beautiful spots. This is Kujuu-san. During the early summer, you can hike up a trail and see a shit load of these purple flowers on one peak, and active volcanic activity, with sulfur and steam coming out, on a twin peak. The country has many beautiful hiking trails, although I noticed that most of them aren't quite as well curated as ours (trails often go directly up without switchbacks), while simultaneously getting way more foot traffic during peak season than they can accommodate.
Like others have said, though, access to most good hiking destinations is by car only. Kujuu actually had tour busses dropping people off, but you really don't want to be hiking when a bus-load of people is all dropped at the trailhead at the same time. Also, imagine rushing back to the trailhead so you don't miss the bus. Sounds awful to me. In fact, that's how I hiked Mt. Fuji and it absolutely was awful.
Additionally, going back to Kujuu, specifically, although it was an easy day hike, there was a campground about halfway up where a few dozen Japanese people had set up tents all in a grassy field. This, like other camping in Japan, was by permit/reservation only. In general, some campgrounds may require booking in advance. If you aren't fluent in the language, you need a Japanese person to navigate it for you, except the majority of them are just as clueless as you are. There really isn't wild/dispersed camping like we have in America, most certainly not out of your car. You'll almost always be right next to other people, albeit Japanese people aren't exactly rowdy campers. Even in the Japanese Alps, you will need to look into staying in the organized campgrounds on the tops of the mountains, which may require reservations. A key detail in anime like Yuru Camp needs to be noted--you will only find relatively empty campgrounds in the off-season, when it's pretty cold.