Ask your hotel about 택배 taek bae or 해주세요 hae joo say yo.
Taekbae is your standard courier. Hanjin or the post office are the main two. There are many others of course. They should be able to schedule a pickup for you and you can prepay.
I used Hanjin to send two big 25 lb bags from ICN airport to my boyfriend's house in Gangnam. Each bag was 10,000₩ and they arrived in two days. Faster options are available, I'm sure.
A hae joo say yo is more like an on demand delivery person. They'll do everything, like go pick up food at restaurants that don't deliver or get your phone from a bar in west Seoul (and you live in east Seoul). Bags might be too big for them as they generally use scooters.