Hokkaido JR Pass - Aomori Station

How much will it cost or can I reach Aomori Station to/from the nearest Hokkaido Station with a Hokkaido JR Pass ?
Is it easy and frequent buses or trains ?

The Hokkaido Rail Pass is valid as far as Nakaoguni Station in northern Aomori Prefecture. Everything south of it is not covered.

Trains from Hokkaido to Aomori do not stop on Honshu until Kanita Station, which is one station or about five kilometers south of Nakaoguni Station.

You have two major options:

Option 1) Stay on board of the limited express train all the way to Aomori. The supplement charge will be around 1500-2000 yen, depending on whether you use a unreserved or reserved seat.

Option 2) You get off at Kanita and use a local train from Kanita to Aomori. The cost is probably going to be a little bit lower, although frankly I don't know how much you will be charged for the short distance from Nakaoguni to Kanita on the limited express. Theoretically, I think you will have to pay a limited express fee for those five kilometers, which may not result in no or only a very small cost difference compared to option 1).