Hi, Doc Mike.
I'm a Japanese working in Tokyo for over a decade.
Let me answer to your questions one by one.
>want to know the best pocket wifi to consider for this rural areas
Firstly, you should check mobile phone operators who provide their networks rather than pocket WiFi routers.
Pocket WiFi routers themselves do not decide easiness to be connected with the internet.
That said, all the three dominant operators (NTT DoCoMo, SoftBank, au) are almost the same on network coverages.
>I'm afarid that the ones that work in tokyo will not work in mountainous regions up north
Generally. network coverages are weak at mountainous areas.
So, I recommend you to check the connectivity on their websites, each providing a map to show the connectivity.
Although they are available only in Japanese,
I just put links to network compatibility maps offered by DoComo, SoftBank, and au.
If you want to check it through the links below by yourself, put Japanese word (like "秋田") into their search windows.
If you find difficult to search by yourself,
tell me (or DM to me) the specific area address so that I can check it instead of you.
Lastly, I just put (almost) comprehensive list of Pocket WiFi router plans available in Japan.
With that list, you can easily compare several elements between lots of plans providers offer.
Rental Pocket Wifi (Portable Wifi) Plans in Japanhttps://prepaidsim.visitjapanplaces.com/rentalwifi
Hope you'll have a wonderful experience in Japan.