O'Leary was born in Tralee, County Kerry, both his father (also named Arthur) and uncle Daniel having been talented musicians, too.
[1] When young Arthur's talents were discovered at the age of ten, a visiting barrister, Wyndham Goold became his patron: in May 1844 he sent him to school in Dublin and secured private piano tuition.
With financial backing from others, including John Stanford (father of Charles Villiers Stanford), and with letters of introduction from William Sterndale Bennett, O'Leary was able to study at the conservatory in Leipzig, where he arrived in 1847 to study piano (with Ignaz Moscheles), organ, violin, and harmony (with Julius Rietz).
Between February 1852 and December 1854 O'Leary studied the piano with Cipriani Potter and Sterndale Bennett at the Royal Academy of Music (RAM), London.
Among his more notable pupils were Charles Villiers Stanford, Arthur Sullivan, Stanley Hawley, and Alicia Adelaide Needham.