I do not think it is possible to know based on the application date, but likely to be "around" the anniversary of the first statement date.
Example: applied and approved 1.7.2013. First statement generated 1.25.2013. Recent annual fees have posted on 01.26.2017, 01.26.2016.
(Earlier years' statements are no longer available online., but I can request copies from a list)
This is very helpful as it seems to suggest that the fee actually hits the around the anniversary date of the first statement rather than the date the card was approved.
I have another question. I have not used the $100 airline credit. If I buy an airline gift card, how long does it take for the airline credit to go through? If it in 24 hrs, 48 hrs, and is the credit automatic or does one have to call Amex for a manual credit? This pertains to the PRG card. Thanks.