That reference to Cyprus, there is a substantial Cat rescue centre, near Akrotiri, (you can visit if you're there, but you'll get covered in cats craving a fuss!). If you visit Turkey, masses of cats, as they are well taken care of, even strays.
Cats - as the owners on here will know -- are incredibly nosey, so even if you try and ignore them, you might get some attention. They can also usually sense if people are frightened of them - like dogs I gather -- which of course makes things worse.
So, if you're phobic/wary/frightened/nervous, just remember that TO THEM, you are a huge and potentially threatening human, so their initial response is likely to be based on that? We knew a chap who did deliveries, he said every third house on avg seems to have a dog, comes bounding up to the door, often barking, either happy to see someone, or possibly putting on a show for their owner. He had been wary of dogs, after nipped when younger, so had to train himself to be confident, 'in control' ("Hello dog, good dog") so they know you're not a threat. He DID say that little yappy dogs are far worse than the big ones that folk are often wary of!
Sadly, many cats are above all that, see people as a source of food, a fuss, or sometimes a game, and may or not respond to any moves you make. Also, a subset of cats are quite timid, will look to escape from you, even if they know their owner has visitors. You could, if you have problems, invest in a packet of Dreamies (other treats are available), so if you're in that unfortunate situation, trapped in the vicinity of someone and their cat, just rattle the packet amd you should find you have a small furry and compliant companion, if not a new friend. That's unless they're even more frightened than you, in which case enjoy that feeling of power/superiority?
Oh, PS, as some have said, lots of therapy available, you can do a course to alleviate fears of cats, dogs, spiders, snakes, whatever. I trained myself to be OK with spiders, and now trap them and put them outside, though the big black ones that look to have about 25 legs, and move like a guided missile are still a bit of a struggle. And I overcame a fear of snakes, joined in a session at a zoo, a group of at least 6 were invited to help hold a massive boa (? - I think) and it turned out not to be all horrible and slimy at all, just warm and almost cuddly (!). Sadly I still have an extreme phobia of Piers Morgan on TV, and the slightest sound of The Hollies on the radio (I think for them it was some odd incident in my youth?).