I have to interpret a work of art during modernism. Was this written during the modernism era?
Yes, T.S. Eliot's long poem "The Love Song of J. Alfred Prufrock" was definitely written during the era of modernism. This poem by Eliot, in fact, is an outstanding example of high literary modernism and Eliot is himself often held up at the primary example of this movement or period in literature.
Literary modernism can be defined in many ways, of course, but this poem exemplifies that movement or period in a number of ways. One approach is to look at the poet's complex relationship with literary traditions. (Eliot writes about this topic, of course, in his essay "Tradition and the Individual Talent"). The poem opens with a quotation from Dante, in the original, and uses end rhyme but otherwise seems to reject traditional, fixed forms of poetry. Another approach is to look at the intense interiority of most high literary modernism, including the technique of stream-of-consciousness. A third approach could look at the man's inability to propose marriage as a symptom of the cultural sterility or cultural death that many artists sought to express in their works in the aftermath of the First World War (see, for example, Hemingway's novel The Sun Also Rises or Fitzgerald's novel Tender is the Night.)
Yes, T.S. Eliot's, "The Love Song of J. Alfred Prufrock," is definitely a work of modernism. Eliot himself is often mentioned in handbooks, etc., as an example of modernism. This avant-garde movement, as far as poetry is concerned, is known for moving away from the rational presentation of ideas, and toward "collages of fragmentary images and complex allusions," according to The Concise Oxford Dictionary of Literary Terms.
You will find fragmentary images and complex allusions in abundance in "Prufrock." The speaker starts thoughts and doesn't finish them, speaks of a retreating crab, uses extended animal metaphor/imagery, alludes to Dante and Michelangelo.
Modernism is, at least somewhat, defined by Eliot's poetry, and "Prufrock" is a good example.