Driving distances Japanese Alps

We (4 adults) are planning a trip to Japan in April 2009 for about 2,5 weeks. We will do mostly by train, but have decided to rent a car between Tokyo and Nagoya to visit the Japanese Alps. Can someone give us some advice, if what we have planned is doable keeping in mind the driving distances?

Day 1:
Tokyo - Kamakura - Hakone area
Day 2:
Hakone area - Matsumoto
Day 3:
Matsumoto - Takayama
Day 4:
Takayama - Nagoya

Thank you very much for any assistance! Arrigato gosai-masu! :-)

Day 1:
Tokyo - Kamakura - Hakone area

I advise against hiring a car for Tokyo and Kamakura. It is so much easier to explore by train and on foot. Hakone is also not only easy but attractive by public transportation, although a car can be convenient, too.

How about picking up the car at an outlet in Ofuna or Odawara and start using it for Hakone. Avoid Hakone on a peak season weekend in any case, because it gets too crowded.

Hakone area - Matsumoto

Via Gotemba, Kawaguchiko, Otsuki and Shiojiri following the Higashi-Fujigoko, Chuo and Nagano Expressways. About 3 hours, 4000 yen expressway tolls.

Matsumoto - Takayama

2 hours along National Route, 750 yen for a toll tunnel

Takayama - Nagoya

2 hours, 4000 yen tolls along the Tokai-Hokuriku Expressway

Dear Uji
Thank you very much for your prompt and helpful answer!
Let's say we go by train to Odawara and like to stop in Kamakura. We have our luggage with us. Are there usually any lockers at the train stations? This would also be the case when travelling by train from Kyoto to Hiroshima and we would like to stop for a visit of the Himeji Castle.
Thanks again for you help and best regards from Switzerland!

If you have luggage, then use the car. If you head out of Tokyo in the morning on a weekday to Kamakura, and then to Hakone, and it should be fine. Staying somewhere at Hakone will give you plenty of time to explore.