Japan - Mt. Koya

Mt. Koya (Koya-san)

*This is a beautiful area to explore on foot as there are many historic sites and hidden treasures.  There is also a -5C difference on the mountain from the surrounding Kansai areas which keeps it cool.

 

Okunoin - cemetery of Lanterns path (short course)

 I highly suggest doing this run/walk in either the early morning (dawn/sunrise) or evening (dusk/sunset).  There is both a stone path and gravel road that lead from the downtown shopping area to Kobo Daishi temple/mausoleum.  Ancient stone structures line the path through a cedar forest with various side trails that wind through the woods.  The enchanting thing about this area is that it has a majestic quality to it.  The best times of day to run here are both dawn and dusk because then you are truly able to experience your surroundings.  At first light, the morning mist still hangs in the air and encircles the stone statues which has a spiritual effect.  In the evening, the stone lanterns that line the path are lit which has an enchanting effect.

Koya Town (medium and long course)

Mt. Koya has a small local downtown area with gift shops and restaurants.  I recommend running along the main streets so that you can view the various shrines and temples in the area.  Mt. Koya is known for its Buddhist monasteries, temples, Shinto shrines, and pagodas.  A great way to see many of these sites is to walk or run from point to point.  There are also local buses that operate every 15-20 minutes if you need alternate means of transportation around town.

Chōishi-michi (町石道) 

- Fudozaka Trail, Gokurakubashi Station to Koya (*mountain challenge)

 Although I have not done it, a great challenge would be to run down the mountain to Gokurakubashi station where the cable car begins and back into town.  I'm not sure if this would be recommended by locals as the road is used mainly for buses, but it's worth a shot just to say you've done it:
"A less historic approach, the Fudozaka Trail, leads to Koyasan from Gokurakubashi Station, the lower station of the Koyasan Cablecar. The steep, paved trail is 2.5 kilometers long and takes about an hour to ascend. Considerably less to descend. The trail ends at the Fudozaka-guchi Nyonindo temple hall." (Japan-Guide.com) 

- Kudoyama to Koya (*super mountain challenge)

Another option is to run the entire pilgrim trail from Kudoyama to Koya: 

"The original main approach to Koyasan is along the Koyasan Choishi Michi trail, which begins at Kudoyama Station on the Nankai Koya Line. The trail is marked by stone signposts (choishi) which stand every few hundred meters along the path so that pilgrims can find their way. The markers are numbered (in kanji) in descending order with number one at Koyasan and are formed to represent the five Buddhist elements: earth, water, fire, air and void. The entire Choishi Michi trail is about 23.5 kilometers long and takes around seven hours to hike. The trip may be shortened by two hours and four hours respectively by starting from the alternate trailheads at Kami-Kosawa or Kii-Hosokawa Stations. The hiking trail ends at the Daimon Gate, a two story tall, crimson gate that marks the traditional entrance to Koyasan, but the markers continue on to the Garan." (Japan-Guide.com) 

- Pilgrim Trails (various)

"A trail, called the Women Pilgrims Course, leads around the perimeter of Koyasan from the Fudozaka-guchi Nyonindo past the ruins of the other Nyonindo to the entrance of Okunoin. It is about seven kilometers long and takes about 2.5 hours to hike. 
A more challenging pilgrimage trail is the Kohechi Trail, part of the Kumano Kodo trail network, which connects Koyasan to the sacred shrines of Kumano. Be aware that it is a long and remote mountainous route that takes nearly a week to hike and requires proper planning and equipment to undertake." (Japan-Guide.com)

Japan - Osaka

Osaka

- Osaka Castle area (short and medium course)

If you are looking for some sightseeing during your run, I would suggest heading toward Osaka Castle.  This is also a good place to do some speed training.  Around the perimeter of the castle, on the outer side of the moat, is a park with gravel and asphalt walking/running/biking paths.  During the spring, this entire area is filled with beautiful, pink cherry blossoms.  During the autumn, the Japanese maple trees change color and show-off bright red, orange, and yellow hues.

- Dotombori/America-mura (downtown area)

This is basically the shopping and eating center of Osaka.  With that in mind, plan to run here in the early morning (6-8am) before many people are out and about.  Luckily, Osaka is a "late" city which means that most people stay out late and sleep in.  However, by 8am the morning rush is underway.  I would suggest running on the stone walkway along the canal.  It has a Venice-esque feel to it and allows you to see many of the local hot spots.  If you are up for it, stop for ramen after your run at one of Osaka's best.