Aomori Springs has only one on mountain hotel called the Rockwood hotel with ski in ski out access. The hotel provides a shuttle service to the Aomori airport. You can also stay at ryokans in Ajigasawa Onsen where there is shuttle from the station to Aomori Spring resort in Winter.
Mt Hakkoda has a good selection of accommodation options near the base area ranging from resort hotels to ryokans. Nearby, the famous 1000 man onsen - Sukayu Onsen resort is another place where you stay to access Hakkoda.
Moya Hills has some excellent cabin accommodation on mountain if you would like to spend some time there.
In Aomori city itself, there are lots of hotels. The Shin Aomori station is where the Shinkansen passes, but hotels, restaurants and entertainment is pretty limited there. There are more accommodation options near the JR Aomori station which is a 7 min train ride away. The shuttle bus for Moya Hills and buses to Hakkoda start from here.
