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.
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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).
青森県立美術館
Exhibits about the natural and cultural history of Aomori prefecture.
ねぶたの家ワ・ラッセ
The distinctive red building houses exhibits about the Nebuta, including several floats, if you're unable to make it during festival season.
青森ベイブリッジ Aomori Bei Buridji
This bridge spans a short harbor inlet near Aomori Station and the former ferry port.
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.
八甲田丸
The ferry which connected Hokkaido and Aomori prior to the building of the Seikan Submarine Tunnel is now a somewhat run down museum, but worth checking out if you're interested in regional history, trains, or boats. The rear part of the deck of the ferry serves as a beer garden on summer nights.