Both conflicts saw Russia and the US back opposing sides. However, both were largely unwilling to become heavily involved in the conflict due to the new détente policies and their need to ensure oil supplies.
The significant individuals within the Cold War were at the time of the two conflicts was that under the 6 day war the American leader was Lyndon B Johnson1963-1969, USSR President was Nikita Khrushchev's1955-1964, and the Egyptian President was Gamal Abd el-Nasser 1957 -1970. Nasser was a significant role because he broke away from Arab Policy and stood up to the west.
The leaders in American and USSR during the Yom Kippur war were for America President Nixon 1969-1974 and for the USSR President Leonid Brezhnev 1964- 1982, and the Egyptian President Anwar Sadat 1970- 1981 and the PLO Yasser Arafat 1969-2004. Yasser Arafat is significant because he became the leader of the PLO and was able to create and unite a solid resistance against Israeli forces.
Both the Americans and the Russians used a variety of methods to gain access and control of the resource petroleum. Both sides create tensions within the region so they could divide and rule in order to control the oil. Both side used politics and militarisms and created alliances within the region in order to stop or slow down the other side from gaining total control of the oil. The Russians used propaganda against the Americans utilizing their allegiance to the Israel in order to make other Middle Eastern countries hostile towards the Americans and vice versa.
The long-term result was that America gain control over the Middle Eastern region and the major producers of oil. They formed strong alliances with Saudi Arabia and other Gulf countries that produced petroleum. The success of the Americans can be been today as the price of oil is pegged to the American dollar and not the Russian ruble.