There's no easy way to get to Ephesus from Cyprus and crossing over to No. Cyprus won't make it any easier. Flying means going first to Istanbul or Ankara for a connecting flight to Izmir, which is the closest airport to Ephesus. From Izmir you then will rent a car or hire a taxi to drive you to Ephesus.
It might be marginally faster to fly from So. Cyprus to Rhodes on Cyprus Airways (there's no ferry) where you can take a ferry into Turkey at Marmaris or Fethiye or Bodrum, then a bus north to Ephesus or then take the flight option I've already described.
Expect either method to take you at least one full day, possibly two if you take a bus from So. Turkey to Ephesus. Air/Ferry connections might require you to spend the night in Rhodes before you can catch a ferry to Turkey.
Another possible option would be to fly from Rhodes to Heraklion,Crete and then from Crete to Samos on www.skyexpress.gr. From Samos you then take the ferry across to Kusadasi and a bus or taxi to Ephesus. Later in the season it might be possible to fly directly from Rhodes to Samos if the demand for SkyExpress flights increases.