Born in Belfast, Rice grew up in London where he attended Gifford Street Primary School as a young boy. He later on worked at a greengrocers on Gillespie Road which is the street Highbury Stadium is upon. Veering toward the sport of football, Rice joined Arsenal as an apprentice in 1964. He turned professional in 1966 and worked his way up as a defender through the club's youth and reserve teams. Rice thereafter made his first-team debut in the League Cup against Burnley on 5 December 1967, a match that Arsenal won 2–1.