Don't try to take a bus to downtown: traffic is always bad, more so on a weekend. Sunday is the only real day off, so half the country tries to drive to visit the other half.

Don't despair, there is an easy solution: take the bus from Incheon to Gimpo (the domestic terminal, and the old airport,) and catch the subway from there. That will get you into the city without worrying about traffic.

As to what to do in downtown Seoul, I have no idea. I'm always passing through, and in eight years in country, I've never actually stopped there!

Mark