Your mother's green card allows her to be in the US, and to enter the US, but it has no relevance to Japan. Thus, with an expired Chinese passport and a US green card, she will be denied boarding for the Guam > Japan flight.
She does not have a passport, so it's possible that she will not be able to board the flight to Japan.
A "domestic" flight from Guam to Mainland USA might be possible, but what you're describing is technically an "international" flight to Japan followed by another international flight to USA. If nothing goes wrong, you probably won't have ...
I found the page on Tim's site for the podcast (rather than the transcript) and at the bottom it has a number of bullets that include:
We’re staying at the Araya Totoan
Which feels pretty definitive to me.
Listening to the actual clip, the speaker sounds like he has a good understanding of the pronunciation of Japanese words, so I don't think he mispronounced or misread, but that the person doing the transcript couldn't hear the word.
I believe he said "Araya" (the "ra" sound in Japanese can sometimes sound like a "da" sound to ...
The blog mentions Kanazawa (Ishikawa-ken), so I'm reasonably certain actually incorrect about them staying at Tadaya (多田屋), a rather famous ryokan in Wakura Onsen on the nearby Noto Peninsula, which does indeed feature private outdoor baths:
Update: I was wrong and Kate's answer is correct.
For what it's worth, you don't need to head all the way to Wakura ...
At the current time, what you're suggesting is not possible.
You will not be granted access to enter the country. Your chances of being issued with a visa for the purpose is very simply non-existent.
Even if you were somehow able to enter the country, you would not be allowed use public transport (including taxis) to transit between the two airports.