He studied law at Saint John"s College, Oxford, but switched to art and became a student at Street John"s Wood Art School throughout 1913 and 1914.
Armstrong served with some distinction in the Royal Field Artillery in Salonika during World War One before returning briefly to complete his studies at Street John"s Wood Art School. After a period of some economic hardship, Armstrong began to build a career as a theatre designer in London and to gain a client base for his artworks. He received a commission to decorate a room in the Portman Square home of the art collector Samuel Courtauld, and also painted a frieze for the ballroom at 1 Kensington Palace Gardens.
Armstrong held his first solo exhibition in 1928 at the Leicester Galleries in London.
In 1933 he joined Unit One and displayed a set of semi-abstract paintings at their one, extended, exhibition. Throughout the 1930s Armstrong continued to work as a designer as he also continued to develop his art
He produced a number of remarkable posters for Shell. He worked on a number of Alexander Korda"s film productions, designing costumes for The Private Life of Henry VIII, Rembrandt and I Claudius.
Pro Patria, also from 1938, is Armstrong"s comment on the Spanish Civil War.
During World War Two, Armstrong worked as an official war artist undertaking short contracts for the War Artists" Advisory Committee, WAAC. In paintings such as The Elms and A Farm in Wales he recorded bomb damage to buildings such as churches and cottages. Whilst other artists, including John Piper, praised these works they were criticized by the art critic Eric Newtown in the Sunday Times as being superficial. Several works by Armstrong were included in the WAAC Britain at War exhibition held at the Museum of Modern Art in New York in May 1941.
After the War Armstrong produced a series of large symbolic works including The Storm, The Battle of Propaganda, The Battle of Money and Victory, the last of which imagined the results of a nuclear attack.
The painting was shown at the Royal Academy in 1958 and attracted considerable public attention.
He was elected an associate member of the Royal Academy in 1966 and despite developing Parkinson"s disease he continued to paint and travel until his death in London in 1973.