If you’re in Tōkyō, the first thing to do is going to Ōsaka.
With the Shinkansen you can get to Shin-Ōsaka station, and from there to Nanba station (you can take the
Midōsuji line of Osaka subway and arrive in about 25 minutes);
You can reach Nanba station also from Kansai International Airport or other airports, in case you got to Ōsaka by plane, by using Nankai line.
From Nanba station, reaching mount Kōya by the “Kōya” limited express train heading to Gokurakubashi is the easiest and most comfortable way.
If you don’t want to take the express train (whose ticket costs 760 yen), a cheaper way is going from Nanba station to Hashimoto station. There you can take a local train to Gokurakubashi, where you can get on the cable-car,
passing through the mountain’s forest: this will surely be part of your memories when you will think of mount Kōya!
The cable car carries you to Kōyasan sation: taking a Nankai Rinkan Bus (the ticket costs about 280 yen) and getting off at “Senju-in-bashi” is a good way to enjoy your visit. You can also use your “Surutto Kansai Pass” or your “Kōya-san World Heritage Ticket” to access the site.
With the “Kōya-san World Heritage Ticket”, you can save up to about 800 yen with the return ticket, including also travel on Koyasan buses.
More information here (in Japanese): http://tinyurl.com/llv4yge
Also, be aware that the Japan Rail Pass cannot be used on Nankai railway lines.