Andrews was born in Comber, County Down, the third son of Thomas Andrews, flax spinner, of Ardara, Comber, and his wife, Eliza, daughter of James Alexander Pirrie and Eliza Swan and sister of William Pirrie, 1st Viscount Pirrie. He was a great-grandson of the United Irishman leader William Drennan. He was educated at the Royal Belfast Academical Institution, and then at Stephen's Green School, Dublin. At Trinity College Dublin, he had a distinguished career: he became a senior exhibitioner (1897) and a prizeman in civil and international law (1898), and graduated in 1899 with honours in ethics and logic. He was also a gold medallist and auditor of the College Historical Society. A sports fan Andrews had passions for shooting, golf, cricket and sailing (mainly on Strangford Lough).
