Lit: "castle-north"; the name of the high school at which Narita Yuzuru and Ougi Takaya are 2nd-year students, located in Matsumoto City. Likely fictional. However, the manga implies that the real-life equivalent is Fukashi High (深志高) , which is indeed "north of the (Matsumoto) Castle".
School begins in May. Some of the classes Takaya takes are: Classical Literature, Modern Japanese, English, Math, Physics, P.E. and an art elective with choices of Fine Arts, Music, and Calligraphy. Takaya and Yuzuru both take Fine Arts. Their day is divided into Periods, with one class per Period. It sounds like classes rotate into different Periods as the week progresses; for example, in Volume 2 chapter 4, Chiaki tells Takaya that the Math teacher assigned him a problem for Second Period today, because he wasn't there for First Period yesterday.