The best way to get to Shiroyama Observatory and Shiroyama Kankou Hotel is by catching the Kagoshima City View Bus from Amu Plaza (Kagoshima Chuo Station). Also, the Shiroyama hotel does have a shuttle bus.
The views from this area are unbelievable and on a clear day, one can easily take a picture of Kagoshima City with Sakurajima in the background. Best visibility of Sakurajima from my experience is during cold days.
During the Christmas and New Year’s holidays, the Shiroyama hotel is a pleasure to visit as it is fully decorated with illuminations as seen in the pictures. What better than seeing illuminations with a nice view of Sakurajima!