Do I always need to go to a clinic first or I can do hospital?

Hey guys. Medical system is a bit confusing for me albeit living her for about a year and I still don't quite understand it (didn't had to use it before). I live right next to a big hospital, Jiseikai Hikarigaoka and I'd prefer to go there, but I read a lot and the common consensus is to go to a small clinic first and then you can go to a hospital if they refer you? My wife says she has breath pain, but the closest woman clinic is about 50 minutes away, so hospital is much closer. I have also read somewhere that if you go to a hospital first you have to pay fee for going there first which could be 10k+? Her Japanese is so-so so medical jargon could be a problem.

The only insurance she has is NHI

by zzarGrazz