Do you lose the sub though at the same time? I'm guessing you must as it runs off the same amp and the same pre-outs from the HU. It could either be the amp (remote on, or power supply to the amp) or the HU (pre-outs, or intermittant "remote on" to the amp. The amp is very inaccessible. You have to remove the sub which means removing quite a few panels (rear speaker panel, boot panel, boot lip panel). You can buy a replacement sub on ebay for about £30, but if it's the HU then this won't help and will take a few hours work to fit.
When it cuts out, you need to fade the HU to the rears and turn up the volume to max and listen to see if you can hear any hiss from the rear speakers (you'll need to get your ear right up against them). If you get hiss then the amp must be OK
I'd wait for the fault to occur, then disconnect the sub (not as easy as it sounds, you need small hands to get to the red connector at the back of the rear light cluster).
Then check the remote on is working with a 12v testing screwdriver or multimeter. The pin outs of the connector are shown below. You might be able to get to the right pin without unplugging the connector. If the remote on is OK, check the permanent live. If you wiggle the loom around does it make any difference.
If both of these are good, it's either a faulty amp, maybe a connection inside the amp, or duff pre-outs from the HU.
Best place for a changer is ebay, either genuine OEM unit (13 pin connector) or Panasonic DP9060 but you'll need a CD bus adapter (about £30). Also check you have the bus cable already installed, it's in the rear left cubbyhole behind the seat belt reel, not all A3/S3's have them.
