The ticket is not cheap - 16,000Y (160USD) one way, and you usually need to come back, but cheaper than a domestic plane. For people on tourist visas the JR East 'flexible four day pass'
http://www.jreast.co.jp/e/eastpass/ is good value. Any 4 days of unlimited JR East travel within a one month period for Y20,000 yen.
JR has been heavily promoting Aomori as a tourist destination with the slogan My First Aomori.... A direct translation of the Japanese Hatsu Aomori. Hatsu is a useful expression to mean 'first', it can be first ever or first of the year. E.g. first love = hatsu koi, hatsu yuki = first snow of the year, hatsu mode = first visit to the temple for the year, hatsu mono = the first time to eat a seasonal food for the year (the food is the mono, not the experience). First time to Aomori
I was a bit surprised to see the posters of the rail staff at the stations on the extension of the route. All have their full names written with the station they work at.... I can't imagine any business getting away with that in Aus. Aomori is much prettier than the bland, sterile pictures suggest.