Armstrong was educated at Mowden Hall School in Stocksfield, Northumberland and St Mary's Music School in Edinburgh, where he was a chorister at St Mary's Episcopal Cathedral from the ages of 11 to 13. He attended Durham School and Trinity College, Cambridge on music scholarships. He played the piano – which has been alluded to in several "The Armstrong and Miller Show" sketches – and the cello, the latter of which he dropped in favour of the "much more masculine" oboe.