Hakone or The Japanese Alps?

My boyfriend and I are going to Japan in April 2nd-12th. We arrive on the 3rd evening and leave on the 12th evening therefore have only 9 nights and 9 full days there. It's our first time there.

We wanted to spend some days in Tokyo and Kyoto but also a couple of days in between somewhere in nature. We did some research and it appears that either Hakone or the Japanese Alps could be a good option, although I'm not sure whether the Alps will be open? We wanted to stay in a traditional Ryokan (with Onsen if possible) with amazing food in a quaint village with nice nearby nature walks on those 2 nights. We would prefer a more traditional and authentic Japanese location then a very touristy and commercial one, as I'm a photographer and prefer to go to places which are off-the-beaten-path. Which location would you suggest in this case - Hakone or the Alps? If the Alps, where in the Alps? Would it be too far for us to go only for 2 nights? Or would you have an even better suggestion of a location for a 2-night nature escape? Would you have a nice Ryokan preferably with an Onsen to recommend?

Many thanks in advance for your help, much appreciated.

Indeed, the higher elevations of the Alps are still covered by snow, but the medium and lower elevations are not anymore.

My recommendation for you is Okuhida in the Northern Alps:
http://www.japan-guide.com/e/e5940.html

And within Okuhida, I feel you may enjoy Fukuji Onsen.

There may be some snow left, and I am not sure whether there are many walking trails available. But it feels traditional and authentic.

Hakone is very touristy, and its ryokan and baths are overpriced. On the other hand, it is easy and cheap to get to...

Thank you so much for your brilliant suggestion, the location and onsen look beautiful and we have managed to book a night there on the 7th of April and a second one (8th of April) at the Maruhaci Ryokan in Maze. Is Maze and this Ryokan beautiful and authentic too? We read that the food is delicious.

Secondly, we are struggling to find accommodation in Kyoto for 3 nights from the 9th-12th of April 2015, where would you suggest that we look? We actually managed to find a place for the first 2 nights (9th-11th) but need to find a place for our last night (11th). We could also sleep somewhere near Kyoto on our last night, somewhere maybe which is on our way back to Tokyo, as our flight departs from Tokyo on the 12th evening. Could Nara be a good option? Or Mount Koya? or Himeji? Any suggestion, would be most welcome. Thanks in advance! Carolina

Nara is awesome, one of my favorite places in Japan and easy to get to from Kyoto. You could try Uji as well which is really pretty. Another place is Hikone which is a nice little castle town, and a short drive from Maibara the first shinkansen stop out of Kyoto.

Try looking for hotels in Osaka as well; it's a short train ride from Kyoto, and often has more availability. It's also got the better nightlife, and is a fun city in its own right!

Otsu is also good for Kyoto. There are lots of places between Osaka and Kyoto, like Takatsuki and Ibaraki that are worth checking. Places nobody has heard of with hotels, like Joyo City may also still have vacancies.

Thanks everyone for the great suggestions, we have been super lucky and managed to get a booking at the superb Yoshikawa Inn in Kyoto, we're so happy!