Jongmyo was created in 1394.

Jongmyo happened in 1995.

It depends entirely on what you like, but you should first take into account that you should spend the last 2.5 hours getting back to the Airport and getting through security, especially if you are going to Incheon International (as opposed to Gimpo, which is a little closer to the city center). You should also take into account which hours and which days you are going to check what may be open or not.

If you like historical buildings, I would recommend going to Gyeongbokgung, which is the most impressive of Seoul's five palaces and the National Palace Museum there. This will probably take around 3 hours to see. If you wish to see the other palaces and/or Jongmyo Shrine, you are well-positioned to see them as well, but you may not have enough time.

If you like shipping and restaurants, I would recommend visiting Gangnam district which has lots of shopping and an incredible variety of Korean restaurants. (You can find restaurants anywhere in Seoul, but the more trendy ones are in Gangnam). This is also the namesake district of the famous Psy song "Gangnam Style" so that may be excitement in and of itself.

If you like Buddhist-style contemplation, Bongeunsa Buddhist Temple is almost always open for visiting and sometimes open for foreign visitor prayer.

If you like walking around outside, I would recommend Namsan Park, which encircles Namsan (the Southern Mountain). There is also Namsan Hanok Village in the park which is a recreation of what a Korean village would have looked like centuries ago.

If you want foreign (Non-Korean) food and entertainment, I would recommend Itaewon district, which is full of wonderful restaurants. (I personally like the Jordanian one, but my friends love the American grillhouses.)

If you are looking for accommodations, pretty much every district in Seoul is safe (there is practically no crime at all) and you should probably choose somewhere close to the district where you want to spend the night before going to sleep or close to Seoul Station to take the train back to the airport.

Here, I'll give you 10. I would choose the Changdeokgung Palace.

Changdeokgung Palace - one of the five grand palaces of Joseon-era Seoul. Literally meaning "Palace of Prospering Virtue", the complex is comprised of 13 buildings and 28 pavilions. Changdeokgung is at its most beautiful during the fall, when leaves turn colorful and start to fall in a gentle, gradual shower.

N Seoul Tower - a 236.7-meter communications tower built on top of a mountain in Seoul, enabling it to reach an elevation of 236.7 meters above sea level. The tower features 4 observation decks, allowing visitors to see much of the surrounding city for a fee. The 4th and highest deck is a revolving restaurant which makes a complete rotation every 48 minutes. Visitors must take a cable car up the mountain to be able to climb the tower. A café and gift shop are also available. The N Tower is especially endearing to couples due to one of its balconies, where they can hang locks with romantic messages and throw away the key, symbolizing the permanence of their love for one another.

Namdaemun Gate - this historic structure was once part of an ancient city wall. The gate, also called "the Great Southern Gate", suffered major damage from a fire back in 2008. The site is currently full of ongoing construction. Nearby Namdaemun Market acts as the wholesale center for South Korea, where gift items, accessories, textile, appliances and other items are for sale.

Admiral Yi-Sun Shin Statue - a monument to the 16th century Korean maritime hero who designed the turtle boat known as geobukseon, used in combating Japanese naval vessels during his time. The statue presides over Gwanghwamun Square along Sejongno Boulevard. A sunken plaza is being constructed under a water fountain in front of the statue, connecting the subway station and a ground-level plaza.

63 Building - this 249-meter building is the 3rd tallest in South Korea. 63 refers to the total number of floors: 60 are above-ground while 3 are devoted to basement levels. 63 Building has an IMAX theater, an aquarium, convention center, banquet hall, lots of stores, and an observation deck called the 63 Golden Tower. The building is located at the southern tip of Yeouido Park, a long and narrow island provided with nature areas, bike paths, and walking trails.

Songjeong Beach - Located in Pusan, this 2-kilometer long beach is made up of fine sand and weathered seashells. Songjeong is the most northerly of Pusan's beaches, and is open every year between July 1st to August 31st. Although it is not as commercialized as Haeundae or Gwangalli Beach, it also does not possess the same level of facilities.

Jongmyo Shrine - the oldest and most authentic of the Confucian royal shrines, dedicated to the memorial services of dead kings and queens of the Joseon Dynasty. Jongmyo is listed among UNESCO's World Heritage sites and is also the longest Korean building of traditional design. The south entrance gate was reserved for spirits, the east gate was for the king, and the west gate was for performers of the sacred ritual.

Bongeunsa Temple - just north of COEX Mall and Convention Center, this Buddhist temple contains a number of buildings and statues. Bongeunsa is one of Korea's major temples, and has undergone many repairs and reconstruction. The complex also has a "Temple Stay Program" wherein visitors can experience the life of a monk for a few hours.

Chongdong Theater - near Deoksugung Palace is this small, modern-style building which houses the performance hall at the basement to shield theater-goers from the cold winter. The stage itself is very big in proportion to the hall, making the use of microphones unnecessary. Concerts in Chongdong tend to highlight the best of Korean music and arts, making it ideal for foreigners who do not have enough time to experience the length and breadth of traditional music.

Mount Pukhansan - a mass of granite near Seoul, a favorite nature trail for Koreans who wish to escape the stresses of urban life. Those who get to climb up Baekundae, its tallest peak, is treated to a 360-degree view of the capital. Historic Pukhan Fortress is within this area, as well as many Buddhist temples.