A bildungsroman is a novel that concentrates on the development or education of a central character. Since "Theme for English B" is a poem, not a novel, it would not be considered a bildungsroman.
Still, the poem is similar in some ways to a bildungsroman because it traces the education and personal development of its narrator.
The poem tells us that the narrator was born in Winston-Salem, where he also "went to school." He also attended school in Durham, and is now a student in the "college on the hill above Harlem" (probably a reference to Columbia University in New York City).
The narrator is the only "colored" (African-American) student in his class, and he lives in a branch of the YMCA, hardly a luxurious lodging. The Harlem neighborhood seems to have a great effect on the narrator:
But I guess I'm what
I feel and see and hear, Harlem, I hear you:
hear you, hear me--we two--you, me, talk on this page.
Although he realizes that what he writes "will not be white," the narrator also sees himself as having much in common with all Americans, regardless of race:
You are white--
yet a part of me, as I am a part of you.
In summary, this poem is similar to a bildungsroman in that it traces a character's education and primary influences.
In the sense that Hughes' experience at Columbia opened his eyes further to the harsh facts of race in America, the poem fulfils one aspect of the bildungsroman; that is, the boy loses naivete and gains worldliness. Otherwise, I don't think it qualifies.