Arapov grew up in Poltava in Ukraine, and received there his first musical instruction. His first desire was to become a pianist. When he moved to Petrograd (formerly St. Petersburg) in 1921, he took piano lessons with Maria Yudina. However, a hand disease later forced him to abandon this dream. His instruction in composition started in 1923 at the Leningrad Conservatory, where he was taught by, amongst others, Vladimir Shcherbachov.
He later became a teacher at the conservatory, and a professor in 1940. In 1951 he became the director of the faculty for orchestration, and of the faculty of composition in 1976. He received, in his time the honours “people’s artist of the USSR” (1976) and Order of Lenin (1986).
Arapov oriented himself first of all towards the officially desirable composition style and worked primarily with nationalist elements, mainly restricting his subject matter to Russian folklore. However, from around 1960, his compositional style started to become more experimental, introducing a more complicated harmonic, rhythmic and sound colour. As subject matter, he more often selected works of literature. Although this later work is generally tonal, the levels of internal discord are higher than previously. In his very last works, Arapov introduced a religious subject matter.