All the places the OP has mentioned are year-round cities with significant local populations*, so there will be no issue generally with finding places that are open, save for the more season-oriented venues like on the beach etc. I'd say the biggest frustration is that hotel outdoor pools are shut - less of an issue in January but can be annoying in the shoulder season when it can actually be very warm.

(*Larnaca's the only one I have no experience of, but suspect there is plenty there too)