Arnstein Rynning Arneberg was born in Fredrikshald, now Halden in Østfold county, Norway. His parents were Mauritz Otto Edward Arneberg (1845–1913) and Hermione Nicoline Mathilde Rynning (1858–1944). Arnstein Arneberg grew up in Lysaker in Oslo. From 1899–1902, he was a student at the Royal Drawing School, now the Norwegian National Academy of Craft and Art Industry in Oslo. Arneberg began his education of the architect with employment as assistant to the architect, Alfred Christian Dahl (1857–1940) in Oslo from 1888–1900. Arneberg studied at the Royal Institute of Technology in Stockholm from 1904 to 1906. He also studied with Swedish architects Isak Gustaf Clason, Gustaf Lindgren, and Erik Lallerstedt. In Stockholm, he studied with a group of Norwegian architect students who came to be influential in the academic environment, including Magnus Poulsson.
Arneberg was one of the first recipients of the Medal of St. Hallvard in 1956. He was named a Commander with Star of the Order of St. Olav and received the King Haakon VII's Jubilee Medal and the King Haakon VII's Commemorative Medal in gold. He was made a commander of the Order of the Polar Star and was elected a member of the Royal Swedish Academy of Arts in Stockholm. The Østfold Architectural Association named the Arnstein Arneberg Prize in his honor. The prize was first awarded for the 50-year anniversary of the ØAF in 2008.