How do airlines pay for aviation fuel and services at an airport they don't normally use? On flights of a regular nature to set destination airports, I assume that there is a system is in place between airlines and airports/providers to pay invoices for fuel/services. 

What do airlines do if diverted to an airport that the airline never normally serves or a freight airline doing a one-off flight to an airport? How do they pay? Does the captain carry a company credit card to pay for these one off charges?
In the United States most places take credit cards.

Large carriers generally get billed, or for smaller airports they use a company credit card. There are even specialized credit cards that are designed specifically for this use.

There is also what is called "contract fuel". Basically there is a company that is a middleman, a fuel supplier who pays the airport fuel company and then rebills the airline. The pilot has a card that identifies them as a client of the fuel contractor.

For example, check out AEG Fuels, a typical fuel contractor. (Please click on that link. I get 25 cents every time you do. Just kidding.)