i recommend going to a place in Myeongdong called "Baekje Samgyetang" -- samgyetang, if you haven't tried it, is a small whole chicken stuffed w/ ginseng and glutinous rice and dates and boiled&served in a cast iron pot and accompanied by a little bit of ginseng liquor -- it's a great healthy sinus-clearing meal and i can't imagine anything better for curing jetlag. To get there, take KAL bus as mentioned, and in Myeongdong the tourist booth can give you a free map of the area and the restaurant is marked on there and the helpful folk in the booth can direct you there...it's on a second floor and has Chinese characters marking it...and big windows where you can peer down on the shoppers on the walking streets below.
If you are in Seoul solo, it's a particularly good option because many Korean restaurants are difficult for the solo diners (some flat out won't serve you, since the tables for groups). But samgyetang is a dish-for-one so you'll have no trouble.
There are no trains from the airport to the city, but you can grab one of the KAL limousine buses into hotels in various parts of the city.
I assume you want to have a Korean meal? If you're in Myeongdong (Seoul City Centre), Mugunghwa at Lotte Hotel is quite nice.
The touristy Itaewon area has an interesting restaurant, Cheongsachorong, which serves imperial Korean set meals (hanjeongsik).
Han-miri in Samsung-dong area also has fabulous imperial Korean fine-dining menus for lunch & dinner.
The cultural area of Insadong also have lots of good restaurants lining the little alleyways off the main street. My fave there is Sanchon, which is 100% vegetarian.
There are about 4 imperial palaces in Seoul, but I think the one you meant is Gyeongbokgung, the largest one. It's got the imperial change of guards in the morning - very colourful ceremony.
There's another imperial palace, Changdeokgung, with beautiful Korean-style gardens & pavilions.
The smallest imperial palace is Deoksugung. It's quite near the Namdaemun market which has a street dedicated to Korean street foods.
Taxis in Seoul are relatively cheap, and they are all equipped with an English translator on call. You can grab a taxi from Central Seoul to Seoul Tower atop Mount Namsan for a panoramic view of the whole city & its surrounding environs. There's a buffet restaurant, Hankook, offering Korean food at the tower.
I loved my lunch at the Noryangin Fish Market..you pick what you want to eat from the vendors and then take your choosings to one of the restaurants on the lower level of the market. We had oysters, baby octopus, a white fish, and shrimp..no English spoken at most of these places but it hardly matters..
Although it is not Korean, the Seoul branch of DTF is fantastic for soup dumplings and all sorts of noodle dishes..
Also, be aware that it takes over an hour from the airport to center city. Buy your bus ticket inside the terminal and they will tell you where to get the bus to your destination outside the terminal.