I went hiking in the Japan Alps, and also stayed in a temple overnight, so maybe I can help.
There are a variety of places to hike. For the Alps, I do recommend Kamikochi, between Matsumoto and Takayama. You can take a bus from either city, hike for several hours, and then get on another bus to complete the trip to the other city. The pictures of the Japan Alps in the link in my sig are from that location.
You can also hike in many, many other places, such as on Shikoku, on Miyajima Island, and to Mt Koya (though you can also take a bus or cable car to the top of My Koya too.)
I don't know of a hiking trail with cabins, but Mt Koya is where I stayed in a temple. There are over 100 Buddhist monasteries on top of the mountain, and one of the oldest cemeteries in all of Japan (and it is a must see!) There are many monasteries that let travelers stay overnight, and prices were reasonable as I recall. Here's the site for it in English, and there's a link at the top for temple lodgings.:
Official website of K Other sites have information as well, just search for Koyasan. It is easy to get to Koyasan, just take the train south on the Nankai-koya line from Namba station in Osaka, you'll be there in less than 2 hours. I lived at the foot of Koyasan, in the city of Hashimoto, for 2 years teaching English.