The 737-300 and 400's I have worked on (many dozens) cannot start the apu using a.c. power.
In normal service they all must have a battery connected and (relatively) serviceable.
The battery is charged using the a.c. via the charger but there is no facility to start directly from a.c. on the classics I have worked on .
It has a d.c. starter only.
The new gens are a different story.