In Kobe, the mountainside area is considered more affluent; while the sea-level neighborhoods are often considered the ‘ low-town.’ (This will not make a difference to you, as the Japanese are prone to discriminating based on hyper-specific cultural properties.). read more
As for “good” neighborhoods, as far as crime, mostly everything is very safe. As far as prestige, Shinsaibashi (a neighborhood close to city center) is considered fairly prestigious and at a certain point ( do not know if this is still the case) was the most expensive real estate in all of Japan. read more