National Health Insurance
You can join National Health Insurance (NHI) at your local city or ward office. NHI is for self-employees (and their families) and students. You are guided to enroll NHI when you make a residential registration. Bring your passport and Zairyu card you receive at the airport. There is no fee charged when you join, they will post-mail the insurance card and premium bills later to your address. City or ward office is typically open 9:00-17:00 Monday through Friday, no appointment is requested.
Social Health Insurance (Company Insurance)
Your company will enroll you with the company insurer on your behalf. Usually it takes a while, can be 2 weeks to a month for you to receive an insurance card. If you need to see a doctor and receive treatment before your card arrives, ask the person in charge of this in your company if they can issue a temporal letter of certificate. If not then you should be able to get reimbursement later from the insurer by presenting the hospital receipt. Japanese hospitals do not reissue receipts in general, please make sure not to discard the receipts. They can also be used for tax return if you had to spend over 100,000 yen for medical care.
If you are not sure which insurance kind to join, ask your employer. Some companies make employees join National Health Insurance.