Born in South Ossetia, Alimbeg Bestayev practiced chidaoba, the Georgian traditional style of wrestling as a youth, but was not introduced to a classical wrestling style until 1951, when his family moved to Vladikavkaz. The breakthrough year for Bestayev as a wrestler was 1955, when he won his only Soviet title in freestyle lightweight and won the freestyle lightweight gold at the World Festival of Youth and Students. In 1956 Bestayev silver in the freestyle lightweight division of the Soviet Championships and gold at the World Cup, and was selected for the Soviet 1956 Olympic team. At the Olympics, Bestayev defeated his first three opponents by fall, but then lost to Gyula Tóth and eventual winner Emam Ali Habibi, settling for a bronze medal. In 1957 Bestayev won the World Championships freestyle lightweight title and, in 1962, won his last medal, again a silver as a freestyle lightweight at the Soviet Championships, to conclude his sporting career. Bestayev then worked as a wrestling coach in Moscow.
Date of birth is usually seen as 6 December 1936, but according to his relatives he was born 15 August 1936.