Alexei Arapov was born on 14 March 1906 in Verkh-Neyvinsky in Perm Governorate in the family of a miner. Arapov graduated from eighth grade. He worked in Yekaterinburg as a stone carver. In 1926, he moved to Donbass, where he became a machinist. He joined the Red Army in 1928. Originally in the 13th Sapper Battalion, he then commanded a machine gun platoon. In 1930, Arapov joined the Communist Party. Arapov graduated from the Ordzhonikidze Infantry School in 1931. He studied at the Frunze Military Academy from June 1940.