He was born at Doneraile, County Cork, elder son of John St Leger (died 1695) by his second wife Aphra, only daughter and heiress of Thomas Harflete of Ash, Frapham and Chilton in Kent.
Arthur's loyalty to his younger brother was evident throughout his life: he lobbied for John's appointment to the Bench in 1714 and was one of the very few peers to speak against a motion to commit him for contempt of the Irish House of Lords in 1719.
St Leger was a Whig in politics, and this, together with the support of influential friends and of his brother Lord Doneraile, was apparently considered sufficient grounds to qualify him for appointment to the Bench.
In fairness to him, it must be said that St Leger and his colleagues faced a daunting task: he himself described the state of the Irish Court of Exchequer as one of "confusion and disorder, almost past remedy".
The appointment of Sir Jeffrey Gilbert as Chief Baron of the Irish Exchequer improved matters for a time: he was a fine lawyer, who after his death gained great fame for his legal treatises, and he was initially very popular in Ireland.
Quite unexpectedly a routine lawsuit, Sherlock v Annesley, involving a dispute between two cousins over the right to possession of certain lands at Naas, County Kildare, which was heard in the Court of Exchequer by Gilbert, St Leger and their colleague John Pocklington, created a major constitutional crisis.
After a short and ill-tempered hearing, the House ordered Black Rod to take the Barons into custody; St Leger's brother Lord Doneraile was one of the very few peers to object to the sentence.
Gilbert and Pocklington were in custody for three months, but St Leger was released early, as he was required to give evidence in a case at the assizes in Cork.