What role does time play in "The Love Song of J. Alfred Prufrock"?

Time plays a crucial role in the poem "The Love Song of J. Alfred Prufrock" by T. S. Eliot. The narrator is a man who is terribly shy and inhibited. He longs for intimacy but does not have the courage to pursue it. Although he attends social gatherings at which women are present, he cannot bring himself to go beyond surface-level pleasantries and expose his real feelings.

The phrase "there will be time" is repeated over and over, but it is obvious that the narrator is deluding himself. He says that "there will be time to prepare a face to meet the faces that you meet," but this is part of his self-delusion. In preparing a face to present socially, he is hiding his true self.

The phrase "there will be time" is used ironically, because for the narrator, in fact, time is running out. He keeps putting off the decision of taking steps to pursue intimacy and a more fulfilling life, and in the meantime, he is growing old. He laments: "I grow old ... I grow old ..." He is worried that people will notice this and remark that "his hair is growing thin" and "his arms and legs are thin." He foresees his own death, at which "the eternal Footman" holds his coat and snickers. In other words, death itself makes fun of him. His self-doubt and depression frighten him and prevent him from taking action to better his situation.

We see, then, that the passage of time is an important theme that runs through the poem. The repetitions of "there will be time" only serve as an excuse to put off what he knows needs to be done. The signs of old age remind him, though, that he does not have an endless amount of time to improve his situation. In the end, in despair, he expresses his belief that he has waited too long and can no longer hope to fulfill his dreams. He imagines mermaids singing, but he does not think that they will sing to him. He believes that he has lingered too long in indecision; now, his time has passed, and it is too late to make his desires come true.

What are the questions in "The Love Song of J. Alfred Prufrock" related to time, action, and death?

Prufrock's comments and questions regarding death are as follows:

"'Do I dare?' and, 'Do I dare?' 
Time to turn back and descend the stair, 
With a bald spot in the middle of my hair—                                
[They will say: 'How his hair is growing thin!'] 
My morning coat, my collar mounting firmly to the chin, 
My necktie rich and modest, but asserted by a simple pin— 
[They will say: 'But how his arms and legs are thin!'] 
Do I dare 
Disturb the universe? 
In a minute there is time 
For decisions and revisions which a minute will reverse."

The above quote (and questions) reflect the speaker's aging, his movement toward death, where he asks if, at this late time of his life, he dares to change things, "disturb the universe"—even while asserting that it only takes one minute to change everything, hoping that it does not matter that it is so "late."

"Though I have seen my head (grown slightly bald) brought in upon a platter" is an allusion to the death of John the Baptist.

"And I have seen the eternal Footman hold my coat, and snicker:" the "eternal Footman" is the personification of death.

And again...

"To roll it toward some overwhelming question, 
To say: 'I am Lazarus, come from the dead, 
Come back to tell you all, I shall tell you all'"

This portion refers to the Biblical account of Lazarus' return from the dead, one of Christ's miracles.

"I am not Prince Hamlet, nor was meant to be" refers to Shakespeare's tragic hero, Hamlet, who dies due to his failure to act.  
"I grow old . . . I grow old . . . / I shall wear the bottoms of my trousers rolled" talks about the passage of time, moving ever closer to death, especially "heard" in the repetition of "I grow old."