Will AT&T sell me a SIM card?

I have an iPhone 5 is broken beyond all reason, but an upgrade next month, My cousin gave me a phone, but the SIM card is a different size than mine, will AT&T sell me a SIM card for my cousins phone and let me use it til my upgrade? Or will it count as my upgrade?

Absolutely!

It'll cost you anywhere between $5 and $35 depends on how bad they rip you off.
But you'll have no problem getting a replacement.

If your cousins phone is a nano sim card you can use the same one, if not it might be a micro then you'll need a new one.

Just walk in with your new phone!

Yes. You can buy sim cards. I'm not sure how much it will cost, I gave my dad my old iPhone 5 and AT&T charged him $20 for the iPhone 5 sim card.

You can use your cousin's phone as a replacement for your broken phone. Make sure that your cousin's phone is indeed a A T&T cell phone. You would have to get another sim card. There's a micro nano sim card (the smaller size) and the regular size sim card. You would have to go to a A T&T store and purchase a new sim card. Because you are using your cousin's phone that won't be consider as a upgrade. So you can still get a new phone next month when you are eligible for an upgrade.