Learn Hardcore Indonesian: Perpustakaan buka setiap pagi.

Indonesian normally drops the present-tense “to be.” Predicates can be adjectives or verbs on their own. Here buka serves as the predicate meaning “open (for business),” so Perpustakaan buka literally reads “Library open.” You would not use adalah here; adalah links two nouns (e.g., Perpustakaan itu adalah bangunan umum).

Is buka a verb or an adjective here? How is it different from membuka?

buka is a base/stative verb that behaves like “to be open.” For hours/availability, use buka intransitively: Toko buka.
membuka is active-transitive “to open (something),” e.g., Petugas membuka pintu. You normally wouldn’t say Perpustakaan membuka by itself; it needs an object (e.g., Perpustakaan membuka layanan baru).

What about terbuka and dibuka—how do they differ from buka?

buka = state/habit of operating: Perpustakaan buka setiap pagi (open for business).
terbuka = “open” as a state (physical/conceptual): Pintu terbuka, or “open to (the public)”: terbuka untuk umum.
dibuka = passive “is opened (by someone)”: Perpustakaan dibuka setiap pagi focuses on the action of opening each morning.

For business hours, buka/tutup is the most natural choice.