I'm not expert in cathodic protection, but does it work with metals under air oxidation? Is it possible to protect a metal from air oxidation by applying an electrode to it? Or it has to be inside an electrolytic solution like sea water?
Not according to Wikipedia (current version of that page as of writing this answer) https://en.m.wikipedia.org/wiki/Cathodic_protection
Galvanizing, while using the electrochemical principle of cathodic protection, is not actually cathodic protection. Cathodic protection requires the anode to be separate from the metal surface to be protected, with an ionic connection through the electrolyte and an electron connection through a connecting cable, bolt or similar.
So, the portions of your material exposed to air won't actually be receiving any "cathodic protection" formally per se, though galvanization may still help out quite a bit.