We have been on a Med cruise in early July. Our September one was hot but July was something entirely different. Forty degrees in Rome, 42 in Turkey (Celsius, obviously). Fortunately, I had anticipated this and brought a larger number of short-sleeved and no-sleeved tops than I normally would. Unless I was planning to do laundry (which I don't on vacation), I would definitely not be wearing anything a second time. We changed at least twice a day - once as soon as we got back from our tour and then again for dinner. So be prepared for either more clothing or doing laundry. Few places in Europe have air conditioning other than the buses so it's almost impossible to escape until you're back to the ship. Even on the previous early September cruise, it was still in the low-30's in Rome (80+ Fahrenheit) and high humidity. And with lack of access to occasional AC unless you're on a bus, even the "cooler" temperatures can be hot. This also makes the contrast after dark sometimes a little chilly and no matter what temperature the air is, when that ship is sailing, it can still be breezy and chill on deck. I also find the theatres like the Arctic unless it's full of people producing heat. So, yes, I bring a jacket/fleece/sweater and wrap for dinner.
Regardless of the temperatures, we always wore long pants in port. Europeans aren't much for shorts so you really stand out and you can't go into the churches. If you want to go into a nicer place to eat when in port, long pants - even khakis - just look a little better. I usually have lightweight linen or knit pants or khakis/cotton pants which also prevents my legs from sunburn. Often they are made from UV protective materials so I don't even need to worry about putting sunscreen on my legs. I might wear shorts and capris on the ship but not in port when I don't know when I will be able to get out of the sun. Hats and sunglasses are a must. Dark hair in 80+ F temperatures in a sunny climate is at best uncomfortable and at worst asking for heat stroke. And the comfort of preventing sun in your eyes even if you're wearing sunglasses is not to be overstated. When it's that hot and/or humid and you have sun on your head and in your eyes and you're having to constantly squint can give you a headache in no time flat.
Bring refillable water bottles. Usually
DCL tours will provide a bottle of water either before or after the tour but you really want to have one with you. Some places - museums especially - won't allow you to bring bottles in with you, or if you do, they have to be emptied first. On our Italy trip last year, I bought the bladder style bottles with a carabiner on them. Made it much easier to carry especially as you emptied them and they didn't get in the way when we were in museums. We found lots of places to refill them.