Well this expression can mean I am pregnant and about to give birth very soon. However as your friend is a man, this meaning of "I am due" is unlikely.... 😉
In the situation you give, someone would usually say: I am due (+somewhere). e.g. "I am due at work very soon" - meaning I must leave soon, to go to work. You might also add a time in this context: I'm due at the doctor's at 11 o'clock.