On my cards, the annual fee accrues around the anniversary of the card opening. So it is different for each card. For example, Citi does not assess all annual fees on January 1, or some arbitrary date like that. Since the date is unique to each card, I don't know of any short cut.
To find out the date on each card when the fee will be assessed, just send each bank a secure message and ask them. "When will the annual fee be charged for this card? Thank you."
You will have to enter in the data for yourself whether you choose an Excel spreadsheet or watchdog to manage your cards.