Incheon Airport Railroad is a means of airport link transportation running 11 stations between Incheon Airport station and Seoul station. You can be guided to Incheon Airport Railroad and then buy ticket at booth 45 of arrival floor or travel center of Incheon Airport Railroad (a basement level between a passenger terminal and a traffic center) or customer information center of Incheon Airport station.
When you arrive in Jeju International Airprot, you can use Airport Limousine Bus to go to Lotte Hotel Jeju.