Alas, I wasn't able to find much to do in Nagoya the one time I was there for a short visit.

I did enjoy Meiji Mura in nearby Inuyama, mentioned already by hailstorm.

Even more than this I loved staying in Tsumago and the delightful trek between this mountain village and Magome. Often the advice is to walk the path the other way as it is mostly downhill, but with your fitness levels I can't imagine any issue.

It takes between 48minutes and 64minutes to get from Nagoya to Nakatsugawa (the longer time is if you take a cheaper train, in other words a ¥2,740 fare vs a ¥1,590 fare)
From there it's a short cheap bus ride -or longish hike- to Magome (or go to Nagiso station and get the bus -or shortish hike- to Tsumago).
If you look up Tsumago & Magome and/or the Kiso Valley (AKA Kisoji) on the web or in this forum you'll see plenty about the area.
Click here for a picture preview.