North (N) Seoul Tower: Korea's Space Needle at Night
There's also a teddy bear museum at Seoul Tower, and a few historically significant points in the vicinity.
To get to the tower at the top of Namsan Mountain, you can take a shuttle bus from Myeong-dong Station exit 3, Chungmuro Station exit 2, Seoul Station exit 9, Itaewon Station exit 4, or Hangangjin Station exit 2. It costs 950 won in cash, but it's cheaper if you use a transportation card and transfers are available.
From Exit 3 of Myeong-dong Station, you can also walk a ways to a cable car (8500 won round trip, 6000 won one way); this route includes using a free slanted elevator that is pretty slow. The shuttle buses are cheaper and faster. If you really want to walk up/down the mountain, it looks like the shortest route is from Seoul Station, though finding good details online is pretty tough.
North Seoul Tower is open daily, 10:00-23:00 (10:00-24:00 Fridays and Saturdays). The six restaurants open by 11:00 and close at 23:00 (22:00 for the snack bar). The observatory costs 9000 won ($9) for adults.