You need to have a tax ID number, which for a natural person usually requires residence permit or proof of residence, and documents proving you aren't Cypriot. They focus mainly on your family ties. You might be asked to show your parents' passports, birth certificates, death certificates, and other documents like that.
Once you have that, you can get the non-dom certificate remotely through a lawyer or accountants, if you don't feel like taking a second trip to the tax office.