How do airlines pay for aviation fuel and services at an airport they don't normally use?
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.)