Hi there,
Thanks for the kind words! Lets take a look at your base JR costs to see if a 21 day pass will be more economical than purchasing individual tickets:
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Narita airport terminal 1 > Tokyo: ¥3140, (62mins, 79km)
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Tokyo > Otsuki: ¥2860, (85mins, 87km)
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Otsuki > Kyoto: ¥14650, (235mins, 561km)
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Kyoto > Hiroshima: ¥10990, (112mins, 380km)
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Hiroshima > Kyoto: ¥10990, (126mins, 380km)
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Kyoto > Toyama: ¥9420, (162mins, 284km)
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Toyama > Niigata: ¥10350, (212mins, 336km)
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Niigata > Tokyo: ¥10270, (129mins, 333km)
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Tokyo > Narita airport terminal 1: ¥3140, (58mins, 79km)
JR ticket cost: ¥75810 for 9 legs
- 21 day rail pass (¥57,700) saving: ¥18110
As you can see you'll make a minimum saving of ¥18,110 with a 21 day pass in addition to retaining flexibility. Because of this I can certainly recommend the 21 day pass option. You also won't have to worry about doing day trips if you fancy for the 21 day period.
Here are some more thoughts on your itinerary:
Mt Fuji (Kawaguchiko)
For Mt Fuji you should head to Kawaguchiko. The JR Pass will take you as far as Otsuki station. From here you'll need to use the non-JR Fujikyu lines up to Kawaguchiko, which wil be around ¥1110 each way.
Use of the JR Pass in Tokyo
You can get great use of the JR Pass around Tokyo and visit many of the sight seeing spots. Here's a map of JR lines in Tokyo for reference. I didn't include this in your calculation so you could make much more savings by using the rail pass here.
Further research
You can look up precise train timetables and costs at the Hyperdia travel planning website and learn how to use Hyperdia for JR Pass holders through our video blog.
Hope this helps!