Meena is very arrogant and class-conscious; she refuses to acknowledge Sunil because she feels he is a man _______________.

In this sentence, Meena is described as arrogant and class-conscious, meaning she looks down on others she considers socially beneath her.She refuses to acknowledge Sunil because she feels he is not important.So, the phrase "a man of no importance" fits perfectly with her attitude. The best choice in this case is option B. 

The other options either convey the opposite meaning or imply that Sunil has some level of importance, which contradicts Meena’s attitude, so they are not suitable.