I'd probably say "A bulb just burned out" without "got." If the person I was speaking to and I both know exactly which bulb - perhaps are sitting next to a lamp and the room just became dark - I'd use the definite article, "The bulb just burned out."
The version with "A bulb" would suit the case where, for example, we are in the kitchen and we see that the dining room just became less well lit (but not really dark). We know that the dining room has a ceiling fixture with four bulbs. We could assume one of them burned out, but we can't know which of the four bulbs burned out until we go into the dining room to look.