J. Alfred Prufrock is an anti-hero because of his looks, his lack of self-confidence, and his own metaphors about what he is and is not. While a hero is expected to be handsome, Prufrock describes himself as distinctly unattractive: He has "a bald spot in the middle of my hair," his "arms and legs are thin," he is growing old, and he anticipates a time coming soon when he will "wear the bottom of my trousers rolled."
The poem is replete with examples of his "indecisions" and unanswered questions that suggest his lack of confidence in his ability to communicate and charm others. He envisions that others will pin him out like a specimen and watch him wriggle. He asks himself whether he dares, and whether if he asks the "overwhelming question," whether it would be "worth it after all." He imagines that his date would reject what he asks or what he has to say by "turning toward the window." He states, "It is impossible to say just what I mean," showing he has no trust in his ability to properly communicate his deepest thoughts.
Finally, Prufrock provides three negative metaphors showing he is an anti-hero rather than a traditional hero. He states, "I am no prophet." Even though he understands some deep truths about life and death and aging that the plastic culture around him seems content to ignore, he does not have the ability to change minds and hearts like a prophet would. He says, "I am not Prince Hamlet," casting himself as an extra, one who, rather than the leading man, might even be "the Fool" in a Shakespearean drama. Finally, although he has "heard the mermaids singing," he doubts "that they will sing to me." He is not an epic hero, like Odysseus, or a fairy tale Prince Charming, who can commune with other-worldly romantic creatures.
As to the "overwhelming question" Prufrock holds within himself and never asks, it could be a relationship question for the woman he is meeting. However, it could be a metaphysical question. Prufrock, despite his anti-heroic qualities, has the ability to see beneath the facade of his society to view the pain, aimlessness, and squandered opportunities of humanity. He has seen his own "greatness flicker," and he understands the weight of mortality. His question would "squeeze the universe into a ball," suggesting it is a heavy, momentous question containing a revelation as great as the one that could have been told by Lazarus, whom the rich man wanted to send back from Hell to warn his brothers. Thus it could have been more on the order of, "Why are we here?" or "What is the meaning of life?" Such a question certainly would have been a turn-off to his date, who was typical of the people who "come and go, talking of Michelangelo."
Although Prufrock is definitely an anti-hero, his question was nevertheless certainly worth listening to and trying to answer.