An organized tour would be easier if you don’t know the city. However, if you get on the Arakawa or Tamagawa rivers, they have cycle paths that go a long way out of Tokyo. Eventually you’ll get to the mountains🙂
I don’t know much about MTB in Japan but there are others on here who might be able to help you. There’s Fujimi Panorama MTB park but that doesn’t open until the end of the month. Looks amazing from the vids on YouTube but doesn’t help you this week. I see a few MTBs out around Ome (western Tokyo) but I’m not sure where they ride.