Tokyo - Yamanouchi (Shibu Onsen) - Matsumoto - Kanazawa - Osaka - Koyasan - Kyoto - Nyutou Onsen - Hirosaki - Tokyo

Hi,
Well, if you couldn't make a 21 day rail pass pay off, nobody could.
Although one thing that pops out is that it would take you at least 6½ to 7 hours or more to get from Kyoto all the way to Nyuto Onsen. It would make more sense and be less tiring if you didn't make such a long trip in one stretch, and put most of your Tokyo stay in the middle of your trip.

The pass can take you to nearly all the places you want to go; a few exceptions are to Shibu Onsen where you can take JR up to Nagano Stn and Koyasan which is not covered by JR (a Koyasan World Heritage Ticket can save you some money).
Since you are already seeing Tokyo, you can largely skip Osaka during the day. After the temples of Kyoto close down around 5PM, you can zip over to Osaka for the evening. At night the city comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg, and Abeno Harukas Bldg are wonderful.
On another evening, you could also go and see Kobe.
Instead, I suggest you take at least half a day and see Nara - it was also a former capital and has some of Japan's best sights. Nara is often neglected by some tourists, which is a shame. Missing the Todaiji Great Buddha would be a tragedy. Nara Park has a lot of other great places, such as Kasuga Grand Shrine, Kofukuji Temple, the Isuien and Yoshikien Gardens, and feeding the many deer in the area.

For most people, 3-4 days for Tokyo are quite enough. You might consider a day trip to someplace nearby, like Nikko, Kamakura/Enoshima, Yokohama, Takao, the Hitachi Seaside Park, or even Sendai. If you go to the Hitachi Seaside Park, you could also see the Kairakuen Garden in Mito and your timing might be very good to see the plum blossoms come out, which is the garden's best time of the year.
You could also see a lot of cherry blossoms in Japan, especially (on average) from the end of March through the first part of April. You can see a list of popular viewing places here.

Matsumoto has a wonderful castle, but not much else that really stands out. You might look at spending more time in Nagano City.

You can see regular rail fares, routes and schedules on Hyperdia.
Under "More Options", be sure to uncheck the Nozomi box since you can't ride that train using the pass.

It looks like you'll be busy, but you will have an unforgettable trip.

Best of luck.