Kumano Nachi Taisha is in the southern part of Wakayama Prefecture and getting there takes a bit of time. A daily trip from a distance from Nagoya or around Osaka may be good.
First, take a shinkansen to Nagoya Station. In Nagoya take the “Nanki” Limited Express train and get off at the last stop, the Kii-Katsuura Station.
It should take approximately 6 hours to get there from Tokyo, and this journey is fully covered by the Japan Rail Pass.
First from Tennōji Station in the southern part of the Osaka Loop Line take the “Kuroshio” Limited Express train to Kii-Katsuura Station. It takes around three and a half hours.
All the route above described is covered by the Japan Rail Pass. However, note that the pass doesn’t allow you to ride on the “Nozomi” fastest Shinkansen, and this means that the journey will take you some 20-30 more minutes to reach Nagoya from Tokyo.
A direct flight is available from the Tokyo Haneda airport to Kii-Shirahama Airport.
A ticket for the local line costs 30,000 yen but if you buy it in advance it is possible in some case to receive a discount (sometimes as cheap as 10,000 JPY).
From Kii-Shirahama Airport take a Meiko bus to the Shirahama Station.
(Timetables for the Meiko bus, in Japanese: http://tinyurl.com/kl5mxny)
From Shirahama Station take a train to Kii-Katsuura Station.
From Kii-Katsuura Station
From the station, take a Kumano Kotsu bus to Mount Nachi. There is roughly one bus every hour.
(Timetables for the Kumano Kotsu bus, in Japanese: http://tinyurl.com/lhg67qv)
I think it’s convenient to visit the Kumano area by car too. However, access from the Osaka/Nagoya area where there is no highway will probably take time.
Lodging and renting a car in the Kii peninsula is also suggested. By the way, we stayed at Shionomisaki, the southernmost point of Hoshu Island!