If you ever need to find a doctor in Japan who can communicate in English, the US Embassy in Japan maintains a list of clinics/hospitals with doctors who can communicate in English.
Because we are tourists without Japanese health insurance, we pay the complete cost and possibly get reimbursed by the travel insurance or regular insurance. (For example, my current Cigna plan reimburses for emergency or urgent care). Tourists are also sometimes asked to pay upfront. Most places take card if over a minimum amount (JP¥10,000 seems to be a common minimum) or cash. It is becoming more common to be charged an additional fee (I saw one that was 50% of the normal fee.) to visit a hospital without a referral.
Although some clinics or hospitals claim to have English-speaking doctors, the other staff may not be fluent in English. That means some amount of Japanese may be needed to successfully make an appointment. Like in the US, some offices have an online system for making an appointment, so immediate browser translation is handy! Some offices require calling, and that is where you may need some minimal level of Japanese.
Making an appointment by phone means calling from the hotel line or something like Skype. You would make the appointment the same way you make any other appointment.
Making an appointment online is also straightforward, except that the form may ask you for the furigana of your name. When a situation calls for that, I feed my name into a an English-to-furigana translator and copy and paste it. The furigana is so they know how to pronounce your name, and they can correct the record after you arrive if it really is wrong.
Like in many countries, you may encounter reluctance to treat people who are not of the same nationality. You can read more than a handful of anecdotes from people rejected on sight from Japanese hospitals. You will just have to try a different location or try to get a Japanese-speaker who ideally also is ethnically Japanese to accompany you.
I personally have not had any issues when seeking healthcare in Japan yet.
I think you are less likely to have issues with the facilities on the embassy list.
