If you have the Apple Card already added to Wallet, you don't need to do anything for a new iPhone. Just sign in with iCloud.
If you have a new iPhone and have not paired the metal credit card, if you have the original packaging all you need to do is bring them close, like AirPods.
Here's an example:
(0:45 seconds in)
If you don't have the packaging you can do it manually through the Wallet app.