For Tokyo, two hikes I'd recommend are Mitake-san and Takao-san. Both are day trips out west from central Tokyo.
http://www.japan-guide.com/e/e3036.html
http://www.japan-guide.com/e/e3029.html
Takao-san will likely be more crowded. Right now until mid-October they have an all you can eat + all you can drink special going on at a beer garden at the summit.
Oh and both should be accessible by JR Pass. For Takao-san you'll have to switch to a non-JR train for one stop.