Driving car on Kyushu

We're going to Beppu on Kyushu by train. Going to Mt. Aso would be interesting and, after reading many posts here and on other sites, car must be most convenient.
But to my actual question. How is it to drive from Beppu to Kagoshima along the east coast?
It is something between 2-300 km and the scenary must be splendid. Is it heavy and slow traffic or nice empty roads? How long time can it take (without stops)?

It is a nice drive along the east coast, but very long. So, if you have limited time, I recommend to drive from Beppu to Aso and then take the expressway along the west coast to Kagoshima. Travel along the west coast is much faster.

I would take the Yamanami Highway from Beppu to Aso. If the weather is good, you'll have some good views of Aso from the side of Mt. Kuju. If you want you can stop off at Kurokawa Onsen along the way.

From Aso, the fastest way to Kagoshima is to take Route 57 toward Kumamoto City. You have quite a bit of urban traffic, though, before you get to the toll road entrance. There is a faster way by taking the road south from Otsu City, under the airport, and getting on the highway ramp at Mashiki. Heading south to Kagoshima is about 170 km and a Y4000 toll.
From Yatsushiro, you head through (and under) the Kyushu Sanchi Highlands, a pretty remote area. However, I think there are 23 tunnels between Yatsushiro and Hitoyoshi, the longest of which is over 6 km. Another long tunnel after Hitoyoshi brings you into the Sendai River valley and Ebino City. The volcanic complex of Ebinokogen and Kirishima National Park is due south of the city. The highway skirts the west side of Kirishima and takes you down to Kagoshima.

Route 10 down the east coast of Kyushu is not that interesting and it takes forever. If you do go this way, don't miss Saitobaru Kofun just to the NW of Miyazaki City.