To get some real hiking you just have to take the train two hours from Shinjuku. Here you can chose between many trails that connect different parts of the trains and busses around the mountain massive. The peak of Tanzawa-san is 1567 meter but there are several over 1600 meter in the area. Here are many huts in the mountains but also a lot of people hiking so booking in advance is recommended. For a day tour I recommend to climb To-no-take south of Tanzawa-san. Take the Odakyuu line from Shinjuku to Shibusawa station from where buses are running up to Ookura every 30 minutes. Here you are at the foot of the mountain and have some steep climbing in the beginning. I had some problem to get on the right trail from the beginning but then it is like a motorway with steps and wide trails. There are a lot of wildlife and birds so take binoculars with you.