No, he did not convert to Islam. This is a rumour that has been going around for years now and it has been disclaimed on several websites who have verified it by either speaking with Mr. Armstrong or his secretary.
Quote [Zafar Bangash]:
"The obsession with celebrities is not new. The most fantastic story was spun around Neil Armstrong, the first man to set foot on the moon, beginning in early 1983. We at Crescent International received phone-calls and letters from Muslims around the world inquirying about Armstrong's conversion. The story was imaginative and rather fascinating: after returning from his moon journey in July 1969, Armstrong went on a world tour. It was claimed that when he arrived in Cairo, the Egyptian capital, he heard the adhan being called out. Intrigued by the sound, he inquired about it and was told that this was the Muslim call to prayer. He is reported to have said that he had heard the same sound when he landed on the moon. This led him to embrace Islam! When this story reached fever pitch, and we were inundated with calls and letters from around the world, we contacted Armstrong's office in Ohio. This writer personally spoke to his secretary, who revealed that although Mr Armstrong had great respect for Islam, he had not become a Muslim and that the adhan story was pure fiction. She also said that the many calls and letters from Muslims from different parts of the world had intrigued her. She in fact offered to send a letter confirming our conversation and even requested that we publicise the facts to put the affair to rest.
Armstrong's secretary did write to us confirming that her boss had not embraced Islam. This, however, was not enough for some Muslims. Later still, some journalists arranged a conference call through the US embassy in Delhi to talk to Neil Armstrong personally and to hear from the horse's mouth, so to speak, that he had not become a Muslim. Only then did the story come to some sort of an end, perhaps much to the disappointment of the hero-worshipping Muslims."