Ok, this is an extremely unlikely scenario, but let's see what we can do. The first real Post- Cold War election was 1992, so it would have to take place in the election of 1992. So to start it off, Bill Clinton still wins the Democratic nomination, and George H.W. Bush still wins the Republican nomination. Here's how it goes different:
Pat Buchanan has just lost the Republican nomination to George H.W. Bush. Bush's lack of interest in campaigning early on led Buchanan and many of his supporters to believe that he had a legitimate chance of winning. Even after losing the primary, Buchanan is determined to give supporters of his brand of conservatism a voice in this election; he decided he will run an independent candidacy for president. He announces at a press conference, "I am running for president because our culture and our country is at stake, and I believe I am the best person to lead America to prosperity." After much consideration, he decides to choose Harvard Professor Samuel Huntington as his running mate. He comments, "Though economically Mr. Huntington is a liberal, he understands the significant importance of Western culture, and I believe his addition to the ticket will bring in a broader range of support." Huntington initially refused, but after considering it and coming to the conclusion that his ticket could not win, he accepted the position to give a larger platform for his ideas.
Shortly after his announcement, however, businessman Ross Perot also announced an independent candidacy for president; he shared many of the economic positions held by Buchanan, and he quickly stole Buchanan's thunder. A Gallup poll conducted in June of 1992 found Perot leading the other three candidates with 39% of the vote, followed by Democratic nominee Bill Clinton, President Bush, and Buchanan in a distant fourth.
To the shock of everyone, however, Perot dropped out of the race, citing Republican threats toward his daughter's wedding. Buchanan announced at a rally that he was who Perot supporters should flock to, "Mr. Perot and I agree on several issues, including an opposition to NAFTA. My candidacy is the logical choice for the supporters of Mr. Perot before his withdrawal." After this, Buchanan shot up in the polls. Perot considered re-entering the race later in the campaign, but decided against it and eventually endorsed Buchanan.
Bill Clinton's campaign had been mired with scandal; he had been accused of an affair and draft-dodging. Shortly before the first presidential debate, however, he was rocked with another allegation of infidelity by a former secretary who worked for him while he was Governor of Arkansas. This caused his poll numbers to dip. At the debate, Buchanan descended upon this like a rabid animal, calling Clinton a "fraud" and "morally bankrupt." Buchanan also went after President Bush, saying, "President Bush promised no new taxes. He lied to your face. We need a real conservative in the White House who will keep his promises, and I am that candidate." After the debate, a poll was conducted, and though most Americans found Buchanan's rhetoric to be too extreme for their liking, they considered him the obvious winner of the debate. At the Vice Presidential debate, Huntington surprised everyone with his extensive knowledge of foreign and economic issues; this stemmed from his lengthy advising to many different regimes around the world. The other Presidential debates echoed Buchanan's similar aggression.
Tragically, shortly before the election, Bill Clinton's running mate, Senator Al Gore, died in a car accident. This slightly boosted Clinton in the polls due to the American people feeling more sympathetic toward him, but the race was still extremely close.
Before the election, a poll was conducted assessing the candidates national support. Clinton lead with 38% of the vote, Bush came in a very narrow second with 37%, and Buchanan came in third with 25%. The results of the election, however, shocked everyone.
Clinton- Electoral Votes: 244, Popular Votes: 39,426,679
Bush- Electoral Votes: 155, Popular Votes: 38,389,135
Buchanan- Electoral Votes: 139, Popular Votes: 25,938,605
None of the candidates won an electoral majority, so the decision was thrown to the House of Representatives. Although Buchanan had come in last in electoral votes and popular votes, he swept the western United States, which has the most states but the smallest concentration of population, along with a few southern states, which allowed him to win the majority of states. Since each state in the House of Representatives only gets one vote in deciding the president regardless of size, Buchanan was elected president due to this Constitutional anomaly. The Senate voted on the Vice President, and since Al Gore had tragically died earlier, the Senate could only vote between Huntington and Quayle, and they narrowly voted to give Buchanan his Vice President.
On December 14, 1992, President-Elect Buchanan was giving a celebratory speech when he was shot by a political activist. The activist had previously called Buchanan a "racist monster" and called Buchanan's victory an "illegitimate abomination." Buchanan was rushed to a nearby hospital, but he ended up succuming to his injuries. Vice President-Elect Samuel Huntington became President-Elect upon Buchanan's death, and President Huntington was inaugurated on January 20th, 1993.
So that is how Samuel Huntington becomes president Post-Cold War. As far as what he'll do in the Middle East, I agree with raharris1973; Huntington believed in cultural wars, not so much wars between counties, so I don't see him getting very involved.