I agree that you should go in thin layers so you can adjust as you go up/down. I just checked the Tateyama Kurobe Alpen Route official guide webpage, and they say for Murodo (the highest point in the route at altitude of 2,450 meters) for mid-August, the average high temperature is 19 degrees C, with the low of 7 degrees C (a day's average is 13 degrees C). This means it will be over 10 degrees C cooler than Tokyo at the highest = coolest point.
(Referenced website - the detailed temperature only in Japanese)
http://www.alpen-route.com/past/index.html