As you might know, Namdaemun Gate (officially called "Sungnyemun") is designated as the "National Treasure No. 1" and owned by the state. As for the state-owned heritage such as the "National Treasure" or the "Treasure". the Cultural Heritage Administration is in charge of their management in general. However, because they are scattered around the peninsula, direct management responsibilities are on the local government authorities. For example, in the case of Sungnyemun, the manager is the Jung-gu District Office of the Seoul Metropolitan Government.
If there is a need to repair or restore a designated cultural heritage, the Administration provides the local authority concerned with financial supports. For your reference, a restoration project is planned for Sungnyemun this year.
Lastly, you can use the searching system on the website to find more details on the designated cultural heritage. Thank you for your interest.