sorry, I was in Limasol about 10 years ago a few times, so I can't give you any names. I can give you a couple of directions, though
it seems like there are meat places and fish places. there are places that will cook up many different types of fish as courses - so you get fish after fish after fish, and they will vary between fried and grilled. really some of the absolute best fish I have ever had. I wish I could tell you the name of the place I went to, but I am sure there are 100 just like it. but aim for having one lunch in a place like that.
there are places that cook a whole sheep in a barrel - not fancy, but fantastic. the mutton is falling apart soft. again, there are a lot of places like this - picture sort of like barbeque - plastic tables with almost nothing but meat on offer.
the other thing that I remember were loukumades - honey puffs. donuts with honey.
I used to buy honey, lokum (turkish delight but don't call it that) and cheese to bring home.
enjoy. wish I could have been more helpful