You're flying Korean Air, right? You must be flying KE or Asiana (or Jeju Air) to check in at Seoul Station.
If you take a taxi to Seoul Station that should be around 6,000W. The AREX Express train (you must take this one, you can't take the all stop train) costs an additional 6,000W. The normal price is 8,000 but if you show your boarding pass you get a discount.