Some copay's can be paid after the visit to the doctors office, urgent care visit or emergency room visit while some places require the copay to be paid when you're leaving the place.
The copay amount is usually a small amount and not a huge expense so it can sometime be easy to cover the cost of the copay after your visit.
When I visit the doctors office I have to pay a copay of $10.00 which is easily payable right after the doctors visit so I never had them bill me.
However if you do not have the money at the time then sometimes the doctors office, urgent care center etc will bill you for the copay.
But it can vary from place to place so it would be a good idea to call up the doctors office and see if the copay can be billed to you.