Stay in a Japanese temple with Buddhist monks
If you are in Japan, one thing you can try as an unusual experience is staying in a shukubo. These are Buddhist temples that host travelers overnight (based on a tradition of allowing religious pilgrims to stay and relax, pray, copy scriptures, and meditate.
As like with Japanese homes, your shoes are left outside the sliding door to your room with a tatami sleeping mat. You can see the monks and share vegetarian food with them. If you decide to go, you should follow the schedule of the temple, and make sure to wear modest clothing as you are in their space as a guest (albeit a paying one).
Also, do not expect wifi, televisions, etc., but plan to meditate or hike or engage in other quiet and reflective activities. It gives you a glimpse of the austere lives of the monks.
In the past, you had to know about these to find them, but now they are listed on booking.com (using that link may get you 10% your first booking), and a specific site for only shukubo — http://eng.shukubo.net/
Mount Koya has a large number of shukubo, but others are scattered around the country, mostly near religous sites.