First, welcome to Japan!

As for accident insurance for bicycling, it’s really common and reasonable in Japan. In fact, we had it for a long time when our kids were students.
However, unfortunately, there doesn't seem to be any insurance with services in English, although there are lots of proper insurance companies that you can apply with easily online, but not in English.
So, I suggest that you should ask for some help from the school or company you plan to work for. I think they will help some with your hard issue in English. Also, they might have some group discount. That's from my experience, because we used to get one from the schools or children's associations which our kids belonged to. They were insured as a group, and the premiums for each person were discounted.

One thing to note is group discounts are usually not available for short-term stays.

For short-term stays, you can apply for bicycle insurance at Seven Elevens in Japan. So, we hope this video helps you. The video shows how to use a self-service terminal to buy insurance. The machine doesn't have English translations, so the video provides a guide through the Japanese process.
https://www.youtube.com/watch?v=ZjYJQ7yjkAc

Here is the website for the insurance sold by 7-11 Japan. Unfortunately, the website is only available in Japanese so you will have to an online translation service.

https://ehokenstore.com/jitensya

Hope you find what you're looking for and this helps you on your search for insurance. Ride safe!