It is completely correct to start a sentence with "an important person to me." When trying to decide whether to use "a" or "an," you need to listen to the sound of the word. If the word starts with a vowel sound, you use "an." That means that you would say "an hour, an offer, an understanding, an owl, an honorable, and an M.A." On the other hand, when the letter "u" makes a "y" sound, or when an "o" makes a "w" sound, you use "a." Here are some examples: a university, a unity candle, a one-way street, a once-in-a-while event, or a U.S. citizen. Sometimes there are differences in British and American pronunciation. Americans pronounce the "h" in historic, so they would write "a historic event."