Old Castle (Kojou) High School is a fictional school set at the site of the castle which was torn down to make way for Katou Kiyomasa's Kumamoto Castle (also named Kumamoto but using different characters—隈本城 instead of 熊本城). It's likely where real-life Kumamoto Prefectural Daiichi (First) High School stands.
It was originally built as a Western school by foreigners during the Meiji Period (Daiichi was built in 1903 as an all-girls school but later become co-ed). The current school was built around 20 years ago (1970s) and is composed of two three-story buildings to north and south connected by a series of hallways with air-conditioned rooms. It also has a sports oval, a prefabricated club storehouse, and a gym under construction. Kumamoto Castle Park is quite close.