While we stayed at a different temple in Koyasan (you often get assigned rather than choose), Yochi-in sounds like one of the prime spots if you can request it. Dating back to 1153, it's got all the elements you want in a temple stay - that perfect Japanese garden for contemplation, morning ceremonies you can join, and chances to try traditional activities like sutra copying.
What sets these Koyasan temple stays apart is how they let you experience the spiritual side of Japan after all the day-trippers leave. Having that secret Buddha Amida Nyorai in the main hall adds another layer of significance to your stay.
Pro tip: When booking temple lodgings in Koyasan, while you might not get to pick exactly which one you stay at, it's worth mentioning Yochi-in as a preference. With room for 60 guests, it's one of the larger options, which might increase your chances of getting in.