If we run out of battery and then charge iPhone, will it turn back on?
When an iPhone is plugged in, it will boot up (that is actually a common techniques used for restarting an iPhone if the power button doesn't work for some reason). 

If the iPhone is powered off while charging, it will remain powered off and will not automatically come back on. 

I understand that you have an iPhone 4 and there is the chance that it was having trouble receiving calls unrelated to the power. However, if the phone was powered off when you checked, that is the simple answer as the cause of the missed calls.