Alexander was the eldest son of Lieutenant Colonel John Robb of the Indian Medical Service and Jean Kirkland Robb. He was born in Poona, India, in 1872.
Alexander was married to Ethel Violet Robb and had two daughters.
Alexander was educated at Aberdeen Grammar School and then at The University of Aberdeen (1889-1891) as a student in Arts. After University, he went to the Royal Military College at Sandhurst. In January 1893, he obtained the Sword of Honour and passed out first in his year. He was then posted to the Durham Light Infantry, 2nd Battalion. On 20th May 1893, whilst serving in India, he went out on the old troopship “Malabar” on her last and record voyage. Those who knew him at this time in the regiment described him as "a fair-haired, blue-eyed, typical British subaltern of the right sort, well-knit and good to look upon". He had plenty of ability and was keen on all games, especially rugby, and was described as "a fine back and very safe and plucky tackler".
In 1897, Alexander was selected for active service with the King's Own Yorkshire Light Infantry during the Tirah Campaign of the Indian frontier war. At the action of Shin Kamar he behaved 'with conspicuous gallantry' and was mentioned in despatches on 7th June 1898. For this campaign he received the Tirah medal. Alexander was proficient in military sketching and draughtsmanship, and in 1901 he was employed on the Burma Boundary Commission. Shortly after this he was appointed Adjutant of the Burma Volunteers in Rangoon, a post he retained until re-joining his regiment at Fermoy in Ireland in 1907.
After his early active service, Alexander came to Newcastle to perform his term of duty at the Durham Light Infantry Depot, and during this time assisted in the preparation of the University's University Officers' Training Corps candidates for their examinations. He was persuaded to become Adjutant of the corps in September 1912, and was appointed Lecturer in Military History to the University. He was received an MA in 1913. During their time in Newcastle upon Tyne the family were listed as resident at 19 Osborne Avenue.
At Ilkley, in 1913, he was appointed Brigade Major to the combined Officers’ Training Corps Brigade and Adjutant of Durham Light Infantry 2nd Battalion. After leaving here on 5th August 1914, and spending about three weeks with his regiment at Cambridge, he embarked with them at Southampton and proceeded to the Western Front. Alexander's first engagement was the Battle of Aisne. According to the University of Aberdeen's Roll of Honour, whilst leading a counter attack on 20th September 1914 he was severely wounded in a bayonet charge, but continued to lead his men up to around 30 yards from the German trenches. Two men from his battalion picked him up under fire and carried him back to the British trenches, one of whom was wounded and the other killed performing this brave action. Alexander died from his wounds in a hospital at Troyon the same night, aged 42.