iPhone 16 not air printing but shows my printer. How to fix?

It would probably be correct to assume that your printer is a supported AirPrint model. In that case, printing from an iPhone should be possible without an additional app, provided that the printer and the iPhone (and iPad and Mac) are connected to the same network (Wi-Fi router).

Use AirPrint to print from your iPhone or iPad - Apple Support

After double-checking all settings/connections, you may want to try a full power-cycle sequence (do not forget the Wi-Fi router):

• Power off the Wi-Fi router, the printer, the iPhone, iPad, Mac, and all other devices/computers.
• Wait two to three minutes.
• Power on the router.
• Wait two to three minutes.
• Power on the printer.
• Wait a minute or two.
• Power on the iPhone, iPad, Mac, and all other devices/computers.

Also, a VPN app may cause difficulties in some cases. If applicable, try to switch off or remove (at least temporarily).

Which vendor and model printer do you have? (Please be specific about the full model name of the printer too, as some printer vendors can have wildly different printers sharing the same numbers.)