A friend of mine at my hostel here in Seoul needs to get to Gimpo Airport in the morning and is asking what time the subway system commences operation.
While the operation starts from 05:30 to 24:00 approximately, the actual timetable differs among different stations. You can check out each station's timetable at the official subway website, for example in the case of Dongdaemun.
On weekday, First train starts 05:19 (down) or 05:31 (up), and Last train starts 24:49 (down) or 24:58 (up).
Not sure about the morning, but if you catch a subway past 24:00 it will probably stop soon. I've rode on a subway until 00:30 after midnight until it came to its last stop, which was one stop away from where I needed to be.