Armstrong was a deist.
Armstrong was very quiet about his political views, though he might have been closest to a libertarian.
Neil Armstrong was born in Wapakoneta, Ohio. His family moved much throughout Ohio as a result of his father's job, though they ultimately settled back in Wapakoneta when Armstrong was 14-years-old. He died in 2012 in Cincinnati, Ohio from complications associated with heart surgery.
Armstrong grew up in a devout religious family, though I can't find which denomination they adhered to. Regardless, Armstrong reported that they did not force their beliefs on him, rather:
They allowed me to pursue my own interests. They didn't try to dictate to me what I should do or where I should go.
Armstrong's actual religious views are immersed in rumor and conspiracy–much like his famous moon landing. Christians claim Armstrong was a wildly devout Christian, atheists have claimed him as one of their own and some Muslims claim that, while on the moon, Armstrong heard the Ezan (the Muslim call to prayer) and converted to Islam.
Well, let's have a look at the evidence.
When Armstrong applied for a position at NASA, he marked his religious views on the paperwork as "no religious preference." During the first moon landing, Buzz Aldrin, Armstrong's space-travel companion and a devout Presbyterian, held a communion ceremony (a Christian ritual) and reported that Armstrong didn't participate but rather watched quietly and respectfully. And here's the smoking gun, folks:
Aldrin was a lifelong Boy Scout. He loved the Boy Scouts. In the late-1950s, Armstrong decided he wanted to lead a Boy Scout troop, so he went to sign up at a Methodist Church. The paperwork asked the applicant what their religious beliefs were, Armstrong wrote in "Deist." So there you have it Christians, Atheists and Muslims: none of the above.
Politics? No Thanks
Armstrong is considered an American hero. His moonwalk had political motives, of course. It was a show of American scientific, engineering, computing and financial superiority during the Cold War–a message to the Soviets, if you will. Furthermore, it was a part of a trend during the '60s of U.S. government expansion and a nationalistic morale-booster for the American people.
Despite the fame and notoriety Armstrong enjoyed after his space mission, he was widely considered a humble, quiet man who didn't see the point in capitalizing on his achievements. Armstrong was approached by political organizations of all stripes, but he refused their advances and never went in to politics, though many of his contemporaries did.
If anyone could drag his political views out of him, Armstrong seemed to favor a small government approach, something possibly akin to libertarianism. He was unhappy that America seemed to consider itself the "world's police" and spoke of the importance of state's rights.
Let's hear your views in the comments.