The room at Seoul Hostel Center is very small. It doesn't have heater or aircon. If you are not fuzzy and would just book a place to sleep, then you may consider this hostel. It is walking distance to a lot of tourist attractions like Gyeongbokgung palace, Gwanghamun Plaza, Insadong, Cheonggyecheon Stream, Myeongdong, Namdaemun market and a lot more...

After we checked-in at the hostel, we walked back to Jonggak Station. The order of our Seoul City Tour is as follows:

Bosingak Bell Pavilion (보신각) houses the Bosingak. Bosingak is a large bell pavilion on Jongno in Seoul, South Korea. The bell in Bosingak gives Jongno its name, which literally means "bell street." It was originally constructed in 1396 but destroyed many times by both war and fire. It was designated Bosingak by Emperor Gojong in 1895. Under the Joseon Dynasty the bell was rung to announce the time, several times a day and to signal the opening and closing of the city gates as well it was used as a fire alarm. In modern times, the bell is rung only at midnight on New Year's Eve. Because of the massive number of people who attend this ceremony, Metro trains on Line 1 of the Seoul Subway do not stop at Jonggak Station on New Year's Eve. Source: wikipedia. It is at Jonggak Station exit 4, across Jongno Tower (종로 타워)

Jogyesa Temple (조계사) does not give off the solemn and traditional air of the other temples located deep in the mountains, or offer the seasonal scenery of the mountains and the sea, since it is located in the middle of the city. It is easily accessible from Jonggak Station Exit 2, from there walk towards cafe7gram and cross the street towards Dunkin Donuts, walk a little further until you reach this temple.

Ujeongchongguk or "Postal Administration Office of Joseon" is less than hundred meters away from Jogyesa Temple. This might not be known to tourists, we happened to discover it while walking. It is the first post office founded in 1884 in Korea and now serves as a memorial building for post. You can find stamps and documentary records preserved in this building. 

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Gwanghamun Gate (광화문) is the main gate of Gyeongbokgung Palace. Literally it means “may the light of enlightenment blanket the world”, and implies the resounding dedication the people of the Joseon Dynasty had in creating a new dynasty.

Gyeongbokgung Palace (경복궁) is a royal palace located in northern Seoul. It was the main and largest palace of the Five Grand Palaces built by the Joseon Dynasty. The name means "Palace" [Gung] "Greatly Blessed by Heaven" [Gyeongbok]. Admission to this palace allows you to also visit National Palace Museum of Korea (국립고궁박물관) and National Folk Museum of Korea (국립민속박물관).
Gwanghamun Plaza (광화문광장) is a plaza  in front of Gwanghamun Gate. Aside being famous to "City Hunter" drama fans, this place tells more a historical story. Be sure to learn more about King Sejong the Great who developed Hangeul and Admiral Yi Sun-Shin who is a Korean Naval Commander.

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From Gwanghamun Plaza, walk towards City Hall and turn left in between Kyobo Bldg and Gwanghamun Post Office, you will soon see Jongno Tower again. You can take train from Jonggak station to go to Chungmuro Station OR if you still feel like exploring the place, walk toward Bosingak and continue to walk straight until you reach Jongno-3 station. From Chungmuro Station, take yellow bus#2 or bus#5 to Namsan Park.

N Seoul Tower (N 서울타워) is commonly known as the Namsan Tower or Seoul Tower, is a communication and observation tower located on Mt. Namsan and is the highest in Seoul and offers a stunning view of the city. Catch the beautiful light show every 8:30 PM

Here is cost breakdown.