How do I get rid of 'continue syncing data' message when opening Settings on iPhone 16?

That message should not prevent access to settings, tap on the message to show you saw it, then just scroll down.

You can get rid of the message by allowing sync to complete. The best way to do this is to connect the phone to power and Wi-Fi. And the absolute best way is to charge the phone overnight, every night, with either 80% limit set or Optimized Charging enabled.