|image from Kobe Fashion Museum. Note that the museum's layout can change based on the special exhibition content.|
The museum might be a pilgrimage point for fashion designers given their library of fashion-related holdings, but for average people, unless you're already in Kobe, as I was, you probably won't come out here for a visit.
The Kobe Fashion Mart is nearby but has limited shopping as far as clothing goes. It seems like there are several stores for interior designers and a few stores for tailor-made suits. There's an art museum, the Koiso Memorial Museum, which mainly has works by Ryohei Koiso, a local artist. I haven't visited it, but there are some very good reviews online.