A, C, and D are correct, but C is most likely the intended answer.
She told me that her father just went to the post office when I arrived.
This means that when you arrived, the father went to the post office, and did nothing else beforehand. The father did not leave before you arrived, and wasn't even in the process of leaving. He might have left as a result of your arrival. This is grammatical, but the "just" is a little odd, because it implies we should expect the father to do something else as well, which he did not do.
B is incorrect, you can't use the present tense to describe an event that has already happened.
C is probably the intended interpretation. Shortly before you arrived, the father left, and was already gone when you arrived.
D is also fine, it says that when you arrived, the father was already in the process of leaving, maybe he'd just started the car to drive away.