Tokyo in October

Be aware October is in typhoon season, so if you plan hiking or rural activities, keep track of the weather forecast. A trip I took to Japan pre pandemic was interrupted by an October typhoon, I had to alter my plans on the fly to avoid the most heavily hit and flooded area of the country, trains were disrupted, etc.

5 days is a bit tight time-wise.

I highly recommend checking out two areas in the Japanese Alps. You should be able to see foliage at high altitudes.

Kamikochi (breathtaking national park with good hiking) -Takayama (old town)-Shirakawago (quaint village) - Kanazawa - (bonus destination) Tateyama Kurobe (series of cable cars, ropeways, with hiking if time allows)
https://www.japan-guide.com/e/e6040.html
https://www.japan-guide.com/e/e5950.html
https://www.japan-guide.com/e/e5900.html
https://www.japan-guide.com/e/e2167.html

Nakasendo: Tsumago, Magome, ... village towns along an ancient transit route.
https://www.japan-guide.com/ad/nakasendo/

For a mountain experience visit beautiful Takayama,

It's a bit far to get to the Japanese Alps region in 5 days.

Since you said Tokyo, have a look at Mt. Takao. Easy transport from Tokyo https://www.japan-guide.com/e/e3029.html

Another full day trip, or an overnight is the Hakone region. Lots of hiking options and, If good weather, views of Fuji.
https://www.japan-guide.com/e/e5200.html
https://www.japan-guide.com/e/e5206.html

Two other thoughts that are day trips from Tokyo are Kamakura and Nikko. Hiking options. Shrines and temples.

I enjoyed a living history museum place in a park setting close to Tokyo. Many old buildings, farmhouses, samurai house moved to the location. English language signage and docents who were quite helpful. https://www.japan-guide.com/e/e3253.html

I'll be in Hakone (return visit) and Tokyo late October myself. Let's hope for no typhoons!

You have been given some great ideas!

This foliage map should help you make decisions:
https://www.japan-guide.com/e/e2014.html

If you go to Nikko, you will see foliage. It's a bit far from Tokyo and I suggest you stay at an onsen ryokan/hotel for one night for the ultimate Japanese experience.

Overnight in Nikko would be great if you can manage it. If not, it is frequently done as a (long) day trip - long as noted above. There are multiple day tour companies offering bus tours (which may be impacted traffic-wise by fall foliage tourists, also as noted by Barkinpark ) or it can be done by train from Tokyo
https://www.japan-guide.com/e/e3800.html
https://www.japan-guide.com/e/e3805.html
https://tokyocheapo.com/travel/tokyo-to-nikko/

You can check the advice in the above links with a travel app like Jorudan or Navitime, both recommended for transport in Japan. The apps will have up to date info, the links above are from good sources and regularly updated, but might not be absolutely to the moment on any given day.

Don't forget to check my recent trip report. I was in Japan for the cherry blossom season in Mar 2023.