Chemical Kinetics - change to system conditions
As you said, your statement "If you add A and the volume is constant obviously the concentration of A increases" indeed does not apply.  In fact, the number of mols of A never changes.  Add a mol of A, and it reacts irreversibly with B and is gone forever as part of C.  

This means that the total volume must decrease for the concentration to increase, which means that "addition of A at constant T and P" must be correct, as it is the only answer that allows for a decrease in volume and therefore an increase in the concentration of A.  

An assumption made here that is not explicitly stated in the question, is that we never run out of B.