A good starting point for Mt. Koya: http://wikitravel.org/en/Mount_Koya

I don't believe there are any JR services to Mount Koya, even roundabout ones; you'll need to use Nankai for at least the last leg up the mountain. The last hour or so of the trip is incredibly scenic, especially in cherry blossom season, although I'm afraid May will be a wee bit late for that.

You are by no means "required" to attend the morning service (at least not at the two temples I've stayed at?), but it's definitely a part of the experience and you'll miss out if you don't. Zoning out while listening to the priests chant as the incense wafts and random monks clang instruments at intervals is quite hypnotic, which is the whole point.