Muirhead was born at 3 Oriental Place, Brighton, the son of James Patrick Muirhead (1813–1898) and his wife Katharine Elizabeth née Boulton (1819–1890). His father was the biographer of James Watt, the engineer, and his mother was the granddaughter of Watt's partner, Matthew Boulton.
Muirhead never married.
In November 1871, the Royal Engineers were among fifteen teams who entered the inaugural FA Cup competition; after victories over Hitchin (5–0), Hampstead Heathens (3–0) and Crystal Palace (3–0 after a replay), the Engineers met Wanderers, the top amateur club of the day, in the first FA Cup Final, played at Kennington Oval on 16 March 1872, which the Engineers lost 1–0, to a goal from Morton Betts. In the final, Muirhead was involved in the only attack by the Engineers which succeeded in breaking through the Wanderers defence when, according to the match report in The Field, "a fine run by Lieut. Muirhead brought the ball within a few yards of the centre of the posts".
Muirhead graduated from RMA Woolwich and joined the Royal Engineers as a Lieutenant on 2 August 1871. Fellow graduates on the same day were two of his fellow FA Cup finalists, Alfred Goodwyn and Edmond Cotter, as well as Richard Ruck, who played in the 1875 FA Cup Final. Muirhead was promoted to Captain twelve years later.
Between October 1871 and September 1873, Muirhead was based at Chatham, before spending two years in Ireland. In January 1876, Muirhead was posted to Bermuda until November 1878 before a transfer to Gibraltar, returning to England at the end of May 1881, where he was based at Aldershot. Between November 1881 and December 1883, he was based at The Curragh (Ireland), before taking up a post with the Royal Arsenal, Woolwich, where he served until July 1893, reaching the position of "Second Assistant, Building Works, Ordnance Factories".
Between August 1893 and September 1898, he was posted to Esquimalt on Canada's Pacific coast, where he was involved in West coast defence works. He spent the last five years in the Royal Engineers at Pembroke Dock in South Wales.
On 4 April 1902, Muirhead retired on £450 per annum retired pay.