Although my experience at hospitals in other countries is limited, I don’t think the process in Japan is much different. Typically located right by the entrance, you should first look for the front desk called uketsuke. The staff will listen to your health concerns and determine the type of help you need or simply ask you to wait in the lobby. Oftentimes, you will need to take a numbered slip from a machine and wait until your number is called (or appears on the screen).
The staff might even ask you to fill out a form while you’re waiting, such as your name, age, address, any current or previous illnesses, allergies to certain medications, what health concerns brings you to the hospital, etc. Some bigger institutions in downtown areas might offer English assistance and forms, but it’s unlikely at smaller hospitals. There are numerous resources online where you can search for English-speaking medical institutions (see the link at the end).
Like in any other country, it’s always best to make an appointment beforehand instead of showing up at the hospital or clinic because you could end up having to wait a few hours (especially during weekends and peak times).
My best advice for you is to be prepared even before arriving in Japan. Just like you would do with booking your flights and reserving your accommodations beforehand, I would encourage you to also make a list of hospitals or clinics (with staff who speak your native language) in your area of stay. Just an hour or two of research can make a world of a difference if something happens unexpectedly.
English-speaking hospitals in Japan: https://www.alljapanrelocation.com/living-guides/hospitals/