...The guy said it could be the belt not being in sync and it not getting enough voltage to the radio and amp....
Ummmm, accessory drive belts do NOT sync, that makes no sense. Perhaps miscommunication, but it doesn't sound like he knows what he is talking about. A quick inspection of the belt should tell if its the belt or NOT, as well, you'd have lots of belt squeel if it was slipping. Its always possible the pulley's aren't lined up correctly (something would have to be wrong with the alternator out of the box for that to happen), but again, you can tell that by just looking at it at the right angle.
Its just crazy that it was working perfectly fine before.
Did you replace the alternator because it was failing/failed with a new/rebuilt direct replacement?
Did you replace the alternator with an aftermarket alternator that has a much higher current output to power your amp and stereo?
It makes a world of difference? An aftermarket hi-output alternator may have had special instructions for the install and your mechanic may have just swapped it like a direct replacement for OEM.
When you visit the mechanic, have him measure the voltage of the system, it is very possible there is something wrong with the new alternator and it is producing the wrong voltage and your electronics are NOT working correctly because of the incorrect voltage.
The fact the motor is idling poorly also makes me suspect a bad alternator out of the box, its possible low voltage or irregular power supply is making the engine management electronics go nuts, or its even possible the alternator has a bad bearing out of the box and the alternator is failing and startign to seize right away, the resistance from the alternator is dragging down the engine idle.
Take the belt off and spin the alternator by hand, it should spin smooth and freely, if its rough or has resistance, the alternator is bad.