Armstrong was born in Hove, England. He was the son of W. A (clergy) and E. Cripps Armstrong. He married Deborah Wilson in 1933. He has two sons and three daughters. He received a B.A. from the University of Cambridge in 1932 and his M.A. in 1935. His hobbies and other interests included travel and gardening. He was made a fellow of the British Academy in 1970 and a fellow in the American Catholic Philosophical Association.

Armstrong began his teaching career in 1936 at University College, Swansea, Wales. This lasted until 1939 when he moved to the Royal University of Malta in Valletta as a professor of classics. In 1943, he became a classical sixth form master at Beaumont College, Old Windsor, Berkshire, England. Three years later in 1946, he went to Cardiff University as a lecturer in Latin. In 1950, he was named the Gladstone Professor of Greek at University of Liverpool in Liverpool, England. He was made a professor emeritus in 1972.

From 1970 to 1971, Armstrong was named a Killam Senior Fellow at Dalhousie University in Halifax, Nova Scotia, Canada. He also achieved a visiting professorship of classics and philosophy there beginning in 1972. He was also named a visiting professor at Manhattanville College in 1966.

In 1973, he won the Aquinas Medal from the American Catholic Philosophical Association.