Hi,
I'm rated on the A320. As others have alluded to, it sounds like a U/S APU which whilst not uncommon are a fairly normal part of day to day airline operations especially during the summer season where down time for maintenance is squeezed. Each airline has a specific authorised MEL for that aircraft type based on the MMEL (Master minimum equipment list) produced by the manufacturer which lists how long it can remain unserviceable for and what conditions need to be met for dispatch.
Operating without the APU isn't dangerous. It's just a pain and ups the flight crew's workload somewhat. What normally happens is an air start machine or truck is plugged in, an engine is started (normally the number 2 engine) on stand and then the pushback is completed. The tug disconnects and the crew will then perform what is known as a crossbleed start. In essence this uses engine bleed air from the operating engine to start up the other engine. Once both engines are running, it's pretty much ops normal with the main difference being that you only have 2 'generators' (excluding the RAT) so if you lose an engine (highly unlikely), some non essential systems may be shed.
I hope that helps.
Champ