Don't worry about anything else, just go to the entry-exit administration office to fill out the form, pay the money and have a physical examination. After a week, you can get the little yellow book you want, as well as the little red book, some medicines to prevent and treat acute diseases, and a small medicine box, but basically you don't need them. There are few symptoms of death due to acclimatization in South Korea so close, and those medicines are basically not used.
Every big city has an exit-entry administration office. Check online in advance and call to say hello to the time. They usually only have a physical examination in the morning and get a certificate in the afternoon. A week apart.
But in the end, I want to say that with experience, it is not bad at all to enter Korea, because I have to go to the hospital here for another check-up. A little wasteful.