Brunnström was born in Helsingborg, Sweden and was the son of director Fabian Brunnström and Hildur, née Banck. He took a reserve officer exam in 1929 and was a lieutenant in the Scanian Cavalry Regiment (K 2) reserve from 1932 to 1946. Brunnström received a Bachelor of Arts degree in 1932 and Candidate of Law degree from Uppsala University in 1936 before he became an attaché at the Ministry for Foreign Affairs in 1936.