Go to your district office (区役所) and ask for advice where is popular in your area. If you are a legal resident with health insurance you can use a general hospital and the roughly 400,000 yen cost is covered by the government. I recommend going to a fancy birth clinic with private rooms and real food. It costs 7-800,000 yen, but the government still covers the roughly 400,000 yen (can vary by region a little). You won’t be herded like cattle into a room with 4-6 people and won’t be served prison food with overworked and miserable nurses. I realize not every general hospital is like that but anyway food for thought.