Buying a phone
From Akita Wiki
In Japan, a mobile phone is called keitai denwa (ケイタイ電話), or simply keitai. ALTs in rural locations should ask their coworkers or BoE what cell network has good local coverage.
The main providers in Akita are:
Other SIM card options (Article from November 5, 2017)
Rakuten Mobile is also now available in Akita, through the mall in Akita city, or online. There isn't an English service available yet however, so you would need someone with a decent Japanese level to manage. They have a bring-your-own-phone service, and they have a no locked in contract period service as well. They are much cheaper than the Docomo and Softbank equivalent options. If you have a credit card with Rakuten, the process is easier (as you already have an account with Rakuten).
What I need to buy a phone
You need your Residence Card (在留カード; zairyū card). Your BoE will help you get the card upon arrival in Japan, which may take up to 2 weeks.