Rapp was born on 20 April 1905 in Stockholm, Sweden, the son of Johan David Rapp, a wholesaler, and his wife Eva Hedvig Gustava (née Swartz). He passed "studentexamen" at "Saltsjöbadens samskola" in the spring of 1923 and became a sea cadet on 29 June 1923. Rapp took a naval officer exam on 5 October 1926 and became an acting sub-lieutenant ("fänrik") in the Swedish Navy on 7 October 1926. He became a lieutenant there on 6 October 1928 and was then commanded to the Swedish Air Force on 1 November 1928. There Rapp became a lieutenant on 1 July 1930, and he then underwent the Royal Swedish Naval Staff College's general course from 1932 to 1933 and its staff course from 1934 to 1935.