The short answer to the question of what J. Alfred Prufrock does with his time is that he wastes it. He does this in at least two specific ways, both explored extensively in the poem.

In the first place, Prufrock wastes his time in meaningless, boring social rituals involving elaborate etiquette and pointless conversations. The poem begins with a social call Prufrock makes alongside the theoretical addressee of the poem. It is interspersed with the rituals of afternoon tea and polite conversation, leading Prufrock to observe

I have measured out my life with coffee spoons.

The second way in which Prufrock wastes his time is more conceptual. He feels that has momentous things to say, an "overwhelming question" he wishes to pose. But he never takes any opportunity to do this and seems to have wasted his entire life without taking any risks, achieving anything, or asking any truly important questions.

"The Love Song of J. Alfred Prufrock" could arguably have been given the title of another of Eliot's great works, The Waste Land. Indeed, "Prufrock" is a poem all about waste, in which the speaker wastes time, opportunities, and, judging by the perceptiveness with which he describes this waste, his intelligence. The intelligence, of course, is Eliot's, but Eliot may well be describing the type of wasted life he would have been in danger of living himself if he had not been an ambitious poet.