The museum/jomon site are both free, and for that, it's very impressive what they can do. You can go in and view the museum, which houses artifacts from ancient civilization (daggerheads, pottery, etc.) as well as go view the reconstructed Jomon buildings, which is also pretty neat.
Additionally, for a fee, there's a crafts area, where you can work with a teacher to make some sort of crafts. From my understanding, some of the options were a small pottery thing, a stone charm, a weaved charm, although I recall there being ~5 or so options. The different options have different hours, although most of them are available around midday. So, don't come too late if you want to partake in these.
There's also a cafe and some souvenir shops which sell some general Aomori souvenirs, as well as some specific to the museum.
Other than that though, there's not much, and since all of the content is in Japanese (minus what the front desk can give you), I didn't find it very interesting. If you're a fan of ancient history, or perhaps just a bit more imaginative than I, you might enjoy this. Additionally, if you're travelling with children, the site provides a number of hands-on activities, including walking through the housing areas, and a few other things in the museum area.
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