One day tour in Seoul?

1. How long does it take from Incheon Airport to the downtown area? Via subway, bus, taxi, and roughly their costs?

Seoul does not have one particular downtown but for one-day tourists, Myeong-dong and its surrounding areas would be the best choice. It will take about 50 to 75 minutes from the airport to Myeong-dong(the city center). Never take taxi, as many taxi drivers would rip foreigners off. The subway would be the cheapest option which would cost around $4 for a one-way trip. However, you should transfer a line and that can be a hassle for busy travellers. I suggest you to take the Airport Limousine Bus which would cost about $12 and you don't need to change a line. It'll get you to the city center whether it's Myeongdong, Insadong or Gwanghwamun.

2. And more importantly, what should I see? I would like to see more of the modern sections, but places like Insadong, Namdaemun, and Gwanghwamun are also of interest

Insadong and Gwanghwamun are pretty close. You can see Insadong, the Royal Palace in Gwanghwamun and Cheonggye Stream. Namdaemun and Myeongdong can be a bit of walk.

Start from Gwanghwamun where the Royal Palace is. You can walk to Insadong and Cheonggye Stream. If you still have some time, you can go to Myeongdong and do some shopping and take Airport Limousine Bus to Incheon Airport!

http://www.visitseoul.net/

http://www.airport.kr/airport/traffic/bus/busList....

It depends on what transportation you'd take, but the bus would be the best choice :)

Korea is a tiny country so it wouldn't take very long, perhaps an hour or so.

I highly recommend you Myungdong and Namdaemun Market if you're looking for great shopping deals.

But, if you're more into having a tour, you don't want to miss Gyeoungbokgung Palace, a old, beautiful palace of Korea :)

Have a great tour in 대한민국! <3