Depending on your pain threshold I would consider just pushing from the airport in Tokyo straight to Osaka or Kyoto for the first night. You can still catch the shinkansen and be in Osaka a couple hours after landing. It seems wasteful to travel down just for the game and not see the city. This would at least get you the morning to explore the castle or countless other areas. Bypassing Kyoto is a major miss.
Your request for peaceful mountain villages to hike out of does not really narrow the search down, that is like saying you want to go hiking in the Alps but do not want to be near a city. Are you trying to still stay on Honshu? Within a couple hours of Tokyo? I would lean towards Hokkaido in the summer to decrease temperatures and rainfall, northern Honshu or Nagano/Niigata for a fall trip. Nozawa Onsen and the Myoko area have a lot of good hiking opportunities in the fall, so does Hakuba into the northern Alps. Even Hakuba will not be crowded unless you are there over a holiday weekend. Nikko is worth checking out if you want to stay near Tokyo, though the town itself is a tourist hotspot. As soon as you depart the main town and Unesco site you will hardly encounter a soul.
It is also worth considering finishing off your trip in Tokyo for a couple of days and hiking during the day and enjoying the city at night. I did more trail running than hiking while living in the city but you can jump on an early train, head to some obscure trailhead, run/hike for half a day, eat and clean up at an onsen, and then catch a train back into the city before dinner. I was usually with a group so did not have to worry too much about navigation but it was best to travel between two stations so you could cross a lot of terrain. Most of the good stuff is NW, W, and SW of the city.
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