Arthur was the son of Samuel Price, vicar of Kildrought and Straffan in the diocese of Dublin and, from 1672, prebendary of Kildare.
Arthur Price entered Trinity College Dublin, on 2 April 1696, aged seventeen, and was elected a scholar in 1698.
After taking holy orders he was successively curate of St. Werburgh's Church, Dublin, and vicar of Celbridge, Feighcullen, and Ballybraine.
His father's friendship with William "Speaker" Conolly (1662–1729) placed him in the way of the political patronage vital for advancement in the established church at the time.
A few months later Conolly was elected as speaker of the Irish House of Commons, confirming his position as chief undertaker[1] now that the Whigs had returned to power in London.
Price bought the Kildrought town malthouse in 1722 and was said to have placed his land steward Richard Guinness in charge of production of "a brew of a very palatable nature".
[citation needed] Price died in 1752 and was buried beneath the aisle of St Mary's church, Leixlip, County Kildare.