I eventually settled on a temple stay in a Buddhist monastery at a place called Shojoshin-in, in Koyasan or Mount Koya, a religious community and one of Japan’s most sacred sites, and I booked the stay.
I was so happy with the service, I ended up booking both the Koyasan temple lodging and a traditional onsen ryokan, a traditional inn with hot springs.
Take the Nankai Electric Railway from Namba Station in Osaka. This will take you to Gokurakubashi at the base of the mountain (make sure you have your Japan Rail Pass.)
In Gokurakubashi you must take one of the cable cars up the mountain for another 10 minutes. Make sure to get a window seat because it is a beautiful ride.
The cable car will leave you at the cable car station where you can take the bus at the bus station to Okunoin. The bus ride takes 20 minutes and costs 410 yen from the cable car station. The bus stop is at the entrance of the cemetery.
This being Japan, directions are clearly marked and people are very helpful, so you won’t get lost. Either Kyoto or Osaka are convenient bases from which to visit Koyasan.