The short answer
Without Japanese public health insurance, you usually pay the clinic or hospital yourself at the time of treatment and claim the money back from your travel insurer afterwards. Some insurers can arrange direct payment, sometimes called a cashless service, but only when the insurer and the facility agree for that visit. JNTO: travel insurance in Japan
Your insurer can only pay against documents, and Japanese facilities issue them in Japanese, at the facility, usually while you are there. Collecting the right papers before you leave is most of the work.