The newer standby AI is in fact called the ISFD - Integrated Standby Flight Director. It is a mini PFD and is a much better than the old round standby instruments. I seem to recall it has 1 hour's power?
Starting the APU would take too much from the batteries and would involve fooling the a/c into being on the ground. Since the 747 has been operational for 38 years, presumably Boeing's engineers have looked carefully into the pro's and cons of allowing an air start of the APU, and so far decided there is no advantage.