What is the narrator’s occupation? How is his occupation significant?

The narrator works as a math teacher. It is a quite well-paid, respectable job that provides him and his family with a good standard of living, certainly by comparison with most people in the neighborhood. The narrator's job and the stability of his family life allow the reader a way into the story. Baldwin wants to explore the dark underbelly of Harlem life, as epitomized by Sonny's own troubled existence. He knows that most readers of Sonny's Blues will have no idea of what the seamier side of life in Harlem entails. So it's important that we can have a leading character in the story who can act as a guide to this unfamiliar territory.
The narrator's career is important in another sense in that it shows a side to Harlem in which African Americans achieve high academic attainment and status in life through hard work, application, and intelligence. Again, Baldwin is challenging and confronting a persistent prejudice in white society concerning African Americans that many readers of the book will be familiar with.