A very nice place you should definitely visit, as it is near the city center and very close to Terukuni-jinja, one of Kagoshima city's biggest shrine. The climb might take some time and be kinda difficult since Shiroyama Observatory is high, but it is worth the effort : you will find cats, moss, old trees and statues along the way, and catch one or two beautiful views over the city. Once arrived, if you come at the right time, you will be greeted by a volunteer who will be happy to take pictures of you in front of Sakurajima ! This person will most likely not speak any foreign langage, but they were very nice and super happy when I talked japanese to them :) Kagoshima is a small city, far away from the agitation of the capital, still, Shiroyama observatory offers a very welcomed moment of peace and quietness. You will remember your climb and day for sure.