We were in Izmir 5 weeks ago and as you leave the dock area there is a Tourist Taxi stand. It has a large sign listing the fixed prices of various tours. I took a picture of it for just this eventuality, though this site will not let me reproduce it. Never mind. The per taxi cost to Ephesus and including the Virgin Mary's House was 100Euro. I guess you could get 4 to a taxi. Have a nice time. Richard.
No need to do any shoreexcursions - all it needs is a bit of research. If you use a browser and type in the words 'message and cruise and europe' - of the results returned you need approx 3rd link down that begins with the word Europe. Ekol is a tour company that is highly recommended on this site.
As Richard says it is easy to get a taxi to take you to Ephesus, the roads are reasonable. It is about 40 miles away. He should be able to drop you off at the top entrance to Ephesus and pick you up again at the bottom down past the Amphitheatre. Don't forget that you will need to pay for the entrance to Ephesus on top of any taxi price, that price is not negotiable but the taxi fare is. .....Neil.
We just returned from 12 days Eastern Med cruise with Norwegian Jade. It was a FABULOUS trip. We tend toward doing most of the ports ourselves, and hiring private guides if necessary because we enjoy researching the ports and like the flexibility and efficiency of a private tour (we have found it to be cheaper than the ship’s tours as well). For Izmir we contacted with local tour operator: Denizhan at firstname.lastname@example.org . We combined with another family I met on our Roll Call. I join the list of Cruise Critic people that highly recommend him. He was knowledgeable, personable, proud of his country, and completely professional. We started early and were through most of Ephesus by the time the first tour bus arrived. Since it reached 105 degrees that day, we were EXTREMELY grateful to have gotten to see it before the hottest part of the day. Be sure to also see the ruins of St. John’s Basilica. Denizhan also took us inside a local and historic mosque and explained a lot about Islam and Turkish politics. Lunch was the best of the trip. We also visited the museum in Selcuk, which was fine, although it is hard to compete with the Vatican, the Egyptian Museum, and others on this trip. Definitely try apple tea if you can – hot or cold, it’s delicious. All in all, a great stop.