Hi LEM,
I agree, the 737NG APU can be started with either aircraft battery or AC power, with AC power being the preferred when it is available - you'll know it's using AC power to start when the "BATT DISCHARGE" light doesn't come on during the start.
Further, I do hope you had the aircraft battery changed for a fully charged one prior to departure, just in case you had to rely on it for standby power.
If your airplanes have twin batteries fitted, the auxiliary battery is ONLY used during standby power operations, so if one of the batteries is not fully charged, the advertised time available on batteries could be somewhat less than expected, and if you don't carry two batteries, well.......