There is no doubt that the concern Montressor expresses for Fortunado's health is feigned. He wants to make sure that Fortunado has no hesitation in following him deeper and deeper into the cellar. Hie is actually unconcerned about the man's health, for he plans to kill him, but if he lets on that he is uncaring then there is a good chance that Fortunado will just turn around and go back up thereby foiling his carefully planned revenge,
At another level, the feigned concern about Fortunado's health is not all that much different than the feigned friendships that are mocked in this piece - the concept that, in certain levels of society, people will treat you one way to your face and another way behind your back. Poe was no stranger to this as the son of an actress. Although he lost his mother when he was only two, he carried with him the stigma of being the son of an actress - beloved on the stage but, in the eyes of society, not much better than a common prostitute. Many of the people in Poe's life were hypocrites, and this gave him plenty of fodder for writing characters who pretended to be or feel one thing when, in reality, their interior motives were much different and generally far more self-serving.
If you mean was he pretending or was he truly concerned, my opinion is that he was clearly pretending.
I think that someone who is planning to wall another person up in his wine cellar and let him starve to death is not too likely to really be worried about that other person coughing. So why show all the concern?
I think that Montresor is using reverse psychology. He does not want Fortunato to back out of going to see his new amontillado. So he keeps urging him to go home and he keeps saying he'll get the other guy to look at the wine. Both of these are in my opinion just ploys to make sure that Fortunato will really be motivated to go look at the wine.