I agree with Steve. Once your car is running your car and stereo should run off of the altinator, unless your pushing some serious wattage where your current demand is more than your altinator can put out. Your battery is actually another load on the altinator when the car is running.
My first guess would be that your fuel pump might be going bad. I had an old honda that died on me like your car. I was also able to get the car started again. It did this to me several times over the course of a few weeks. Each time took a little longer sitting on the side of the road before the car would start. What I found out was that the fuel pump was overheating and locking up. After it cooled off the car would start and run fine untill it decided to hang up again.
I hope this helps.