When I got back I activated the card and went to my existing citicard account online, where I saw I have a payment due date of 12/13 (Haven't charged anything on the card yet). Does this mean I missed my chance to receive the $250 airline credit this year?
No. Citi actually tells you when next statement is expected to close. After you log on, click on the prestige card to get to account details, then click on the + sign to view more expanded account details.... Just as a FYI, my current due date is Dec 21 and next statement closing date is Dec 23.