How do I print from my iPhone 13 to HP Deskjet F4580?

This printer has been retired according to HP, so most support pages are not available.

lobsterghost1 was probably referring to the HP Smart app for iPhone:

https://apps.apple.com/us/app/hp-smart/id469284907

"For a full list of supported printers, visit: https://support.hp.com/document/ish_2843711-2427128-16?openCLC=true?"

At least earlier, the Deskjet F4580 appears to have been supported, but you may have to carry out a test in order to determine whether the app in question actually will work with your printer.

Typically, you use AirPrint as long as the printer is an AirPrint capable printer. Here is a list from Apple of all AirPrint Capable printers and I didn't see your printer listed --> About AirPrint - Apple Support

If your printer isn't AirPrint capable you'd need to see if the printer offers an App in the App Store, which HP does offer, which may let you print to it from the app directly. You'll need to look at the owners manual for your printer to check.

Should you decide to replace that printer, you will want one with AirPrint support.

Some considerations: Which Printer Should I Buy? - Apple Community

AirPrint (and the related IPP/IPPS, where available) means vendor drivers are often unnecessary.