Hi there,
The JR Pass covers the JR Resort Shirakami, although I do recommend making seat reservations a couple of days in advance. Because full=full. The idea behind this train is of course to experience to ride and enjoy the view of the Japanese coastline pass by.
The Trains stops both a Aomori and Shin-Aomori stations (although the last two trains of the day only go up to Hirosaki), you can get on the Shinkansen back to Tokyo on the latter. Aomori itself is not a very interesting town, most people stay there as stopover before heading to Hokkaido.
Hope this helps,
Daniel