The new Kybele Cafe on the waterfront between Finicoudes and Makenzy Beaches has wi fi too.
I've seen quite a few people sit there working with a cuppa. The owners are friendly and the prices are cheaper than Finicoudes. Places aren't very busy right now - they probably won't mind.
If you prefer to pay and not have to keep buying coffee etc, there is an Internet Cafe in the old town. Near the big Currency Xchange that looks like a castle (just behind the seafront).
State of the art and very classy (it's the one with the obligatory 2 white plastic chairs outside):