If you arrived light-handed, then getting to any major city is not a problem. Traveling by bus in the Republic of Cyprus is very budgetary and comfortable. Bus 425 runs from LarnacaAirport to Larnaca city. One should get off at the stop Foinikoudes and take intercity buses to Limassol, Paphos, Nicosia, or Ayia Napa. You can find the timetable and fares here:
The routes are here: https://m.cyprusbybus.com/