Hi, just saw your post. It was a while ago so hope my reply isn't too late for you!
Promises restaurant is lovely, it's one of my favourites actually. It's upstairs directly opposite fig tree bay. You can either sit inside or on their open balcony. Great choice of food, lovely warm bread to start :-) and I've never been disappointed (eaten there many a time). Well worth a visit!

What sort of food do they serve ?

The restaurant is actually in 2 halves owned by the same people. To the left, is 'Promessa' which specialises in Italien dishes, to the right, 'Promises' which serves international cuisine. (However, I believe you can order from either menu in both) Everything from your Cypriot speciality dish of Kleftico, to steaks, seafood, pasta. Something to suit all tastes I find and great service.