Driving to Kiso Valley

Is this feasible? I'm planning to go on a weekday (15 nov to be exact)
So far I know Narai and Tsumago has parking lots available, not sure about magome.

How much are parking charges like?

Thank you in advance!

How much is fuel costs in Japan? (especially Chubu area)

And how long does it take to drive from Nagoya city to Narai?

A car is the most convenient means of exploring the Kiso Valley, although you can also do it relatively easily by trains and buses.

Is this feasible?

This depends on where you intend to pick up your rental car. If you pick it up inside or close to the Kiso Valley, it would be very feasible. But picking it up far away from the Kiso Valley might not be a good idea.

I'm planning to go on a weekday (15 nov to be exact)

15 November is a Saturday. It is considered a weekend in Japan.

So far I know Narai and Tsumago has parking lots available, not sure about magome.

All three towns have parking lots available. You can see their location in the Kiso Travel Guide on this very website:
http://www.japan-guide.com/e/e6075.html

How much are parking charges like?

If I remember correctly, Tsumago and Narai cost around 300-500 yen, while Magome was free, I think.

Another thing to ask...
How much is fuel costs in Japan? (especially Chubu area)

It has been fluctuating quite a lot recently. They are now around 170 yen.

And how long does it take to drive from Nagoya city to Narai?

About one hour to Nakatsugawa by expressway. Then about two hours by national route.

Thanks Uji!