It is likely to be difficult to offer much advice unless/until you can find copies of the contract(s) you entered into, either with the airline or any intermediary such as booking.com
I've never used booking.com to book flights, but whenever I have booked flights, the confirmation email(s) I have received have provided the terms and conditions, either directly or via a link.
I have found the summary of the terms and conditions which booking.com sent by email but i can’t see anything about payment for check in at the airport
But the email they sent me for check in said “check is now open, and if you don’t check in online you maybe charged