Opened in 1982, the Kobe City Museum combined an archaeological museum and an art museum into one. It's full of historic maps and artifacts related to international trade. Plus, there is lots of Namban (Southern Barbarian) art dating back to the 1500s.
Out front is an opening showing a small bit of history: an old water pipe discovered just beneath the street next to the museum.
Nearby is an old western house that I believe now serves as a tea house or cafe. It's a historical building that has been preserved and repurposed at the same time.
I haven't been able to visit yet, but it is still on my to-do list. It was closed when I was in Kobe, but I wanted to share the few pictures I took while I was passing through.
Kobe City Museum is about 10 minutes from Motomachi and Sannomiya Stations, near Meriken Park, which has a few more touristy things to see and do and a small earthquake memorial. There are also a couple subway stations nearby which might be more convenient based on your approach. It's open 10:00-17:00 (closed Mondays), and admission is only 200 yen. I understand that there is very little English signage.