Will the SIM card from my iPhone 8 fit in my iPhone 13 Pro?

Yes. Both of these iPhones use nano SIMs so the SIM card from your iPhone 8 should fit just fine.

You can read about SIM card support for the iPhone 13 Pro here.

Size wise, yes. But your iPhone 8 Sim card is not one provisioned for 5G service, so you should either use the eSim in your phone or get a new Sim from your carrier.