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Shiko Munakata is famous for his woodblock prints, displayed in this very small gallery along with some of his paintings and calligraphy. Closed Mondays except for national holidays and during Nebuta.
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青森県立美術館
Exhibits about the natural and cultural history of Aomori prefecture.
善知鳥神社
There are nice gardens and ponds tucked away behind the main shrine. There's also a "power spot" -- it's said if you wash your coins in a certain fountain there, you'll be free from money troubles.
アスパム
A large triangular building that can be easily spotted from the train station or the bridge, this is the tourism center for all of Aomori prefecture. It includes a movie about the prefecture (Japanese only) and an observation deck with views of the harbor and city (extra charge).
まちなか温泉 machinaka onsen
If you're tired, weary, and want a bath without going all the way to Asamushi Onsen, you can enjoy a hot spring bath just a few minutes walk from Aomori Station. The small bath outside is nice.