Abortion is not covered by the Japanese health insurance, but is legal in Japan. To find a clinic where abortion is practiced, search for jinko chuuzetsu (人工中絶) and your local area.
To have an abortion, make an appointment at the clinic or hospital first. On the day of the abortion you will be required to bring a certificate stamped by the father of the baby and money in cash.
The certificate can technically be signed by any man, so if the father of the baby is not reachable or refuses to cooperate, ask a male friend to sign. This person will not be contacted.
It typically costs around ¥100,000 to have an abortion and it can be done in one day without staying over at a hospital.