Driving the length of Japan does not sound particularly pleasant, and I say that having hitchhiked from the northeasternmost tip of Hokkaido to Shikoku and Hiroshima. (Not in one go, mind you!) There are some great drives in Japan, but very few of them along the main arteries you'd use to cross the country. Expressway tolls are very expensive, and the regular highways that would be sensible alternatives are slow and full of trucks avoiding the tolls.

Taking the Shinkansen across Japan sounds like a much more comfortable and Japanese way to accomplish the goal, although most Japanese cities are of a muchness.