ICN is located in the west of Seoul, and President Hotel is located at the center of Seoul. Among many feasible modes of transportion, KAL Limousine Bus service is the most convenient.
After getting out of the customs area at the airport, look for a ticketing booth. There are several ticketing booths as marked in the map.
Buy one way ticket to Lotte Hotel/President Hotel (Bus number: 6701). It costs around KW16,000. You can pay cash or with your credit card. (Warning: There are two Lotte Hotels, which are served by two different bus lines. Ask for a ticket to "Sogong-Dong Lotte Hotel" if the clerk is not sure about which Lotte Hotel you want to go. The other one is "Jamsil Lotte Hotel.")
Look for Gate 4B or 11A of the terminal. Just outside of the gate, you will find a few posts with numbers of the bus printed along with the departure schedule. The departure schedule of Bus 6701 from ICN is roughly every 30 minutes from 4:40AM to 10:40PM. Bus 6701 stops at Gate 4B and Gate 11A.
The bus will make several stops. You should get off at Lotte Hotel, which is adjacent to President Hotel. When you get off the bus, the area looks like this. Here is the same area from a different viewpoint. Facing Lotte Hotel, President Hotel is located to your right, approximately 100 meters from the bus stop as shown here. The hotel entrance is like this.