Hi, Bufeo.
Welcome to the Apple Support Communities. We’re glad you decided to reach out about not being able to see your printer in the printer list when attempting to print. If we understand correctly, you were able to add the printer and update the drivers successfully as outlined in these articles:
Add a printer on Mac - Apple Support
Update printer software on Mac - Apple Support
When you attempt to print, you mentioned it doesn’t show up in the list. Are you printing for your Mac or another device on the network?
You mentioned AirPrint. Have you confirmed that your printer is AirPrint enabled?
These suggestions may also help:
Solve printing problems on Mac - Apple Support
All our best.