OK, so I finally made it to Shirakami Mountains in hot-humid early August. I chose to go to the Aqua Green Village area because I can at least enter the buffer zone from there. (I myself don't count in a WHS unless I set my feet within it.) I took a bus from the Hirosaki train station to the village. From the village there are two walking trails: one that goes through beech forest and the other that takes you to three water falls collectively called Anmon no Taki, both of which are in the buffer zone. I combined the two trails and walked on both, which only took me 100 min. It was a nice and rather easy walk. Since I had extra hour at the village, I soaked in a hot spring at the village to cleanse myself off sweat.
I happened to go to Heinrich National Park in Germany just last year, which is also a WHS for its beech trees. Folks at the World Heritage Center at Shirakami told me that the beech trees at Shirakami are supposedly more ancient.