People in Cyprus often spend Saturday night at bars, cafes, pubs, restaurants, and clubs, or simply at home watching TV. In some places, people gather most in the evening, sit, talk, eat, and drink, and pubs or cafes with TVs are common social spots. 1 2 3 4 5
Younger people may go out between about 5–9 pm, when there tend to be more people around, especially in social areas where groups sit and chat. In places like Limassol, people also mention clubs and bars as the main nightlife option. 4 5
If you want the most local-feeling option, try a pub, cafe, or casual bar in the evening rather than only tourist venues. If you want, I can also narrow it down by city like Limassol, Paphos, Larnaca, or Ayia Napa.