"Vacant," generally, means "empty." One special use of it is to suggest an empty mind. "A vacant stare" means that someone is staring out into space, looking at nothing, not paying attention, and apparently not thinking of anything. "An air" can mean an appearance, deportment, behavior. "She had an air of superiority:" she acted as if she thought she were better than everybody else. "She had an cheerful air about her:" everything she did suggested cheerfulness in lots of different ways. I'm not sure, but I think "A vacant air" means "he looked like he was not paying attention to his surrounding." But "A vacant air of listening" is puzzling, because "listening" would mean paying attention to sounds. So I think it means he looks as if he is listening to something, but it does not match any actual sounds. Therefore: He looks as if he were listening to something imaginary in his thoughts.

This sounds like a very poetic use of the language. Because of this, more context would help us understand the meaning! What were the sentences before this phrase?

"He didn't listen at all" means, "He 100% didn't listen." I think the speaker had a message for the man, but he ignored the message. He didn't listen [to the message] at all. I hope that helps. :)