You have not experienced Japan until you have stayed in a temple on Mt. Koya. And I highly recommend Ichijoin as your temple accommodation. Check-in at the temple was fast and easy, dinner served at 5:30 pm in your room, and ours was a Deluxe with Garden view--highly recommend for zen meals and mornings. I feel that the art displayed within Ichijoin was more impressive than the Kongobuji Temple! Plan to spend some time circulating through the rooms here. The vegetarian meals were very filling, you have to enjoy Japanese food, though (not a problem for us). Definitely see the masoleum at night AND during the day, so go after dinner. The onsen baths at Ichijoin--also a must-do, just know your etiquette--are open until 10 pm, and you need to be back for a 9 pm curfew anyhow so you have an hour for the baths. Our room was plenty warm, free fast wifi, we did not care about the TV. Pillows are hard by American standards, so bring a travel pillow if that bothers you. I can't think of a better way to start your day than the 6:30 am meditation & prayer service, truly beautiful. Be sure to wander the room after the service. Breakfast at 7:30 and then we checked out to head back to the masoleum before a 9:30 am cable-to-train back to Osaka. We were in Kyoto by 12:30 pm. Next time, I would stay another night to be able to see a few more sites, hike a little more, and simply spend quiet time on the mountain.
Travel from Osaka: Getting there from Osaka is painless IF you plan ahead and pay a little extra for the fast trains. The Heritage ticket does NOT include the fastest routes, and if you hike you can't use it for the smaller stations. Download the Hyperdia app, which provides train info in English throughout Japan, all lines (and is the best public transport app I have used, I might add). My friend and I hiked the pilgrimage routes, and left our luggage in lockers in Tennoji station so we just had our backpacks--highly recommend this as it was quite freeing in Mt. Koya.
Context for this review: I am a world traveler for business and pleasure, and relatively fit. I'm low-maintenance but understand high quality. I don't review everything I experience on TA. In Japan I have had amazing experiences in Tokyo, Osaka, Kyoto and Miyakojima.