Among Kōya-san's most atmospheric temples, with a thatched roof and an old kitchen with exposed wooden beams, Shōjōshin-in is a wonderful choice for your shukubō experience. Rooms are simple, the cheapest divided only by fusama, but comfortable, and care goes into the meals. It's also the closest shukubō to Oku-no-in.
The nearest bus stop is Ichi-no-hashi-guchi.