6 months long ticket but only stay 1 month

Okay, this might get confusing. My plan is to leave the US and travel to Japan, then to Taiwan, then back to Japan. So all in all, I will arrive and leave from the same airport in Japan. My question is: Can I buy a plane ticket from Januray to July from the U.S. to Japan but only stay there for about a month? (The other five months will be in Taiwan)

In principle, yes. Arrival date from US to Japan in January AND departure from Japan to Taiwan within three months. The airline will be looking at your onward travel from Japan.
As for the round trip ticket back to US, booking that for July is also pretty straight forward on most airline booking sites.
I would simply book all four legs on the same ticket - probably cheaper/easier.

Welcome to Japan. today is black day for us. Whole world feels sorrow,
With our Japanese friends. Not only Japanese society mourn but the entire world.If coronavirus pandemic is not happen then today Tokyo Olympic 2020 is started however waiting for next year with improve conditions.
however my Question: why you stay 1:month instead of 6 months.

Japan Olympic Association take right Decision. Otherwise millions of Japanese affected. This made situation more complicated.