I've just visited Seoul twice. I didn't skate while I was there as it was only for short business trips and there wasn't time. I did look into skating. The best one I found was at the University, although I couldn't find freestyle sessions. It looks to be all public. From what I read on the internet if you're skating out of peak times (eg not evenings / weekends) it should be relatively quiet. I also thought I'd spotted an American living in Seoul whose daughter was skating and taking lessons.
From my understanding ice time there is more geared towards speed skating, so quite possibly the only ice for figure skating is public.
However, I thought Yuna Kim used to train at the University rink (or at least she's registered as a student there).
Good luck in your hunt. If you do find a rink to go skate at let me know and if I have to go over again, I might take my skates next time.
I did find the subway system there really easy to navigate as they have the station signs transliterated into English and they all have station numbers as well as names, and arrows telling you what station is up and down the line. I didn't try the buses although I think you need to get a bus to the university rink.
I was told by work to always carry a sign in Korean with you saying where you're going to and where home is, so if you do ever get lost and need to ask directions you've got something for them to read.
But how exciting to spend a year studying there...