Taxi. You can take a taxi from Gimpo to President Hotel. The taxi stand is just outside of the arrival terminal. KW20,000 should be sufficient for the trip by a "regular" (not "deluxe") taxi from GMP to President Hotel (in front of the City Hall).
Subway. The subway would be the most economical and predictable mode of transportation to move from GMP to President Hotel, although you will have to change the subway once on the way to the hotel.
If you plan to use the subway more than once while in Seoul, buying a "T Money" Card is recommended, which is available at a modest fee in most convenient stores and some subway stations. Otherwise, you need to buy a single use card, whose instruction is here. The machine has English and Japanese instructions.
Gimpo Airport (GMP) is located at the western edge of the city of Seoul, which is served by 3 subway lines (Number 5, 9 and Arex). After getting out of the customs area at the airport, look for the sign to the subway. There are two different ways, which take more or less the same amount of time.
Arex and Number 2 line. Take Arex (to the direction of Seoul Station, away from ICN) and get off Arex at Hongik University Station (홍대입구역) and change to Number 2 (Green) line. Number 2 line is a loop line. Be sure to get on "Inner Loop" toward City Hall (not "Outer Loop"). Get off at "Euljiro 1-ga" (을지로입구역). Lotte Department Store, and Lotte Hotel would be good land marks to follow, as President Hotel is just west of Lotte Hotel. (i.e., Lotte Hotel is between Lotte Department Store and President Hotel)
Arex and Number 1 line. Take Arex (to the direction of Seoul Station, away from ICN) and get off at Seoul Station, which is the last stop of Arex. Take Number 1 subway line to City Hall, which is just one stop from Seoul Station. City Hall Station of Number 1 line is connected to a network of underground shopping center and corridors, which lead to President Hotel. Consult with the map at the station. If you need a visual guidance to move around, first get out of City Hall Station through Exit 5, which leads the (old) City Hall Building. Turning your back to the (old) City Hall Building, (so that you are facing south), you can see Plaza Hotel and Seoul Plaza directly in front of you, President Hotel is a tall yellow ivory colored building in the direction of 10 o'clock (south east of the City Hall).