T. S. Eliot's "The Love Song of J. Alfred Prufrock" is the monologue of a sexually frustrated middle-aged man with unfulfilled desires and an "overwhelming question" (which is never specifically identified). The line in question, "I have measured out my life with coffee spoons," is Prufrock's commentary on the futility of life. He has existed in his current state for far too long, measuring out his life with spoonfuls of sugar in his coffee, with nothing to show for it.
Lines 37-48 reveal Prufrock in his current state: his thinning hair, skinny arms and legs, his coat drawn tightly up around his chin all suggesting his age and decrepitness. But it is the question Prufrock offers in this stanza that is the essence of this poem: "Do I dare / Disturb the universe?" Despite his age, Prufrock still imagines a different life, one the challenges himself and his surroundings. Unfortunately, it is never meant to be as he only wait "till human voices wake us, and we drown."
First, Eliot's "The Love Song of J. Alfred Prufrock" is a poem, rather than a story. That said, I'll answer your question in two parts. I'll let the enotes Study Guide about the poem answer the part of your question that deals with the "coffee spoons" quote:
Prufrock tries to explain why he is indecisive about his feelings toward the woman he is meeting for tea. It is because he knows the kind of social life he is moving toward. He knows how people who live together and have social obligations toward one another act—or are supposed to act. The visual image of the coffee spoons indicates that he himself has had innumerable cups of coffee in unbearable social situations. The aural image of the "voices dying" refers to difficult and embarrassing social conversations that falter while those involved pretend to be listening to music. And so, Prufrock asks himself, how can such a socially inept individual as he is ever hope to assume a part in real human life with this woman?
To answer the second part of your question, I'll quote the lines you ask about and comment in brackets:
And indeed there will be time
To wonder, 'Do I dare? and, 'Do I dare?'
Time to turn back and descend the stair [time to turn around and go back, and if he does, will they see?],
With a bald spot in the middle of my hair [if he turns around they'll see his bald spot]--
(They will say: 'How his hair is growing thin!') [he imagines how they'll talk about him]
My morning coat, my collar mounting firmly to the chin,
My necktie rich and modest, but asserted by a simple pin [he strictly adheres to society's taste in dress]--
(They will say: "But how his arms and legs are thin!")
Do I dare
Disturb the universe [as changing his lifestyle would, which he would have to do if he becomes a part of the woman's social circle]?
In a minute there is time
For decisions and revisions which a minute will reverse [instability of values, what he thinks is most important].
With these explanations and interpretations, you should be able to draw conclusions about what these lines reveal about the speaker's character traits. For instance, his worrying about his bald spot suggests that he is self-conscious and worries about his appearance.