I will iyH be in Yerushalayim for Purim and will be returning to New York where I live on the Wednesday after Purim. When do I need to hear the megillah? Sunday or Monday or both days?

Shalom, Unlike keeping one or two days Yom Tov, what day(s) to celebrate Purim is not determined by the fact that you live in New York. It is determined by where you will be on the morning (sun-rise) of the 14th and 15th of Adar (this year Sunday and Monday mornings). If you will be in Jerusalem on Monday morning (sun-rise) you are obligated to keep Purim on Sunday night and Monday day - hearing the megillah on Sunday night and again on Monday day, and giving mishlo'ach manot, and matanot le'evyonim and having a Purim meal on Monday in the day-time. You may choose to take this opportunity of being in Israel to keep two days of Purim! To do this you need to go on Saturday night (or, to make traveling easier, you could spend all Shabbat) outside of Jerusalem (say to Bet-El, or Tel Aviv) and sleep over there Saturday night. Then you are obligated to celebrate Purim on Saturday night and Sunday (hearing the megillah Saturday night and again on Sunday, and doing all the other mitzvot on Sunday). You then travel back to Jerusalem in time to hear the megillah (again) there on Sunday night and sleep in Jerusalem. Then you keep Purim once again, and on Monday morning hear the megillah (for the 4th time) and do all the other mitzvot (again!!). Of course you are not obligated to do this - but it does sound like a great Purim - after all those years of keeping two days Yom Tov in New York, and one day Purim, you could now do the opposite (na'afochu) and keep two days of Purim!!! If this though sounds a bit to much for you, just sleep in Jerusalem on Saturday night, and then Sunday is not Purim for you, and you will keep Purim with the rest of Jerusalem on Sunday night and Monday day.