A bit late to the party but I have dual citizenship and have been living in Japan on a long term visa for the past 8 years so I know how it works.

It’s really simple. You enter and leave Japan with the passport you used for registration at immigration and the Korean one in Korea. They will ask for departure and entry stamps etc at both passport controls so you just whip out the other passport and show them the stamps. Carry both passports at all times and you’ll be fine.

I also have a small note that declares my dual citizenship on the back of my zairyuu card just to make things easier. I’d recommend you doing it just to be on the safe side. Take both passports and the zairyuu card to immigration and ask them to add the Korean nationality on the back. It only takes 5 minutes or so.