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About half of the over 120 temples in town offer lodging for pilgrims, known as shukubō in Japanese. Prices vary between ¥9,000 and ¥15,000 per night and include two meals. You will be offered the opportunity to join in the morning prayer session, a hypnotic experience involving sutra chanting, incense and gongs. Outside the main season, you can just show up at the Koya cable car station and book from there, but generally reservations are preferred. A full list is available here, but note that not all temples are set up to handle visitors who don't speak Japanese.
Koyasan Onsen Fukuchi-in
The natural hot springs of Koya-san are available only at this hotel, are open 24 hours a day, and strictly for staying guests. Only vegetarian dishes are served here, and the meals, which feature seasonal ingredients, are painstakingly planned and prepared by the head chef with a different course every month.
While the monks don't drink, alcohol is available to guests at dinner, and perhaps even from a vending machine. Temples have set hours at which the front gate is opened and closed, and the time the bath is available. This curfew can be as early as 9 PM, so don't expect to head out after dinner — although you'll want to go to bed early anyway if you want to attend the morning prayers around 5 AM!