Some 757s can draw stby power from both the main and APU batteries. AAs are configured only to draw from the main, hence the 30 minute theoretical limit.
As far as working the problem taking a bulk of that 30 minutes, If you have a failure of the normal powering of the stby buses, it seems wise that you'd start pointing the aircraft towards the closest good airport, while you noted what was working and what wasn't (determine which stby bus was affected) and worked the very short checklist . . . Stby Power Switch - BAT, and called the ground engineers for an additional inputs.
From there you could figure out if you think you could land within the life of the battery, and if not maybe conserving battery by going back to AUTO.