Bubukin started training in aged 12 in the club Krylia Sovetov. He then moved to VVS Moscow, but the team was disbanded in 1952, and Bubukin went to FC Lokomotiv Moscow, where he spent most of his career. He made his debut for USSR on September 6, 1959 in a friendly against Czechoslovakia (he was selected for the 1958 FIFA World Cup squad, but did not play in any games at the tournament). He played in the first ever European Nations' Cup in 1960, which was won by the Soviet team.