This itinerary can be comfortably completed in a week to ten days.
Planning your train route: Japanese Train Route Finder By Jorudan Co.,Ltd. Google Maps can also provide rail information, but currently (March 2014) does not handle multiple destinations for public transit. (That is why the map above has a driving route, rather than a rail route.) You can save money by buying a rail pass before you go to Japan. Although the passes cannot be used on the very fastest trains, the “Nozomi” and “Mizhuho”, they are good on the other high-speed Shinkansen trains.
Hotel Sunroute Plaza Shinjuku is a modern Western-style hotel, very inexpensive (get a double for less than $150 a night), next to a major rail station and the amazing Isetan department store. Be sure to vist the high-end food stores in the lowest level of the Isetan.
Ghibli Museum: a wonderful homage to animation. It is quite small, but plan to spend most of a day to experience it in full. You must purchase tickets far in advance. In the US, you order tickets from the JTB travel agency. It is about a 20 minute train ride from Shinjuku station to Mitaka station, and then another 20 minute walk from the station to the museum.
Ryokan Shimizu: the inn with the true feeling of Kyoto is inexpensive and foreigner-friendly.
There are thousands of historic sites in Kyoto. On your first visit, be sure to take in Kiyomizu-dera Temple, Nijo-jo Castle, and Ryoanji Temple. You can spend any number of days in Kyoto and not begin to exhaust its treasures.
Nara is Japan's ancient capital, a short train ride from Kyoto. Wander the Deer Park and tour Todiaji Temple.
Finally, take the train through the mountains to...
Wander Takayama's delightful streets of historic shops. Shop for all kinds of hand-made goods, foods, and Sake. Stay at least one night.
Plan to spend one or two nights in this ancient mountain village, staying and taking your breakfast and evening meal in a traditional gassho-zukuri style farmhouse. Make your reservations far in advance through Japanese Guest Houses: Ryokan Styles. (This web site also lists guest houses in dozens of other historic villages and cities across Japan.) You will need to take a bus from Takayama to Shrakawa-go; note that there are only a few buses a day.
Finally, take the return bus to Takayama and the train back to Tokyo.