I know you said your dad would want to do a full-day excursion at each port, but do you think you could talk him into doing two multi-hour excursions at a few of the ports, instead of all day? That way you get a break, coming back to the ship to refresh.
Granted in places like Kusadasi, it's somewhat of a long-haul from the port to Ephesus, so two half-day excursions would not make sense there. But, it can be gawd-awful hot there without much shade. In Mykonos, other than walking thru the little town, any excursion would likely be much by bus.....or just enjoy one of the water-side restaurants there and take in the views of the windmills.
So, maybe negotiate your dad to alternate full days with easier days. And, if all is going well, then you can likely add additional shore excursions as the trip goes along or switch from half-days to full-day formats.
Places like the Terrace Houses in Ephesus (if that excursion if offered on Princess) tend to sell out quickly.
In Gibraltar, the cave or siege tunnels will be cooler, and it does not take long to get to the top of the rock (and there should be some breezes). Rome can be brutally warm outside, so maybe opt for Vatican options instead of the Forum and Colesseum (unless early morning touring). A lot of the sights in Istanbul are inside venues, so that port should be relatively easy. Ditto for Barcelona, just avoid Park Guell (which can be very sunny and hot, crowded, etc. that time of year).
The Acropolis in Athens will be likely brutally hot. We were there one year at the very end of August before catching a cruise. I remember after touring the museum (which is lovely and cool), seeing a young lady wringing out her shirt in the ladies room and almost melting after going to the top of the Acropolis (so we opted not to do that....regret it to this day, but it would have been way too hot).
But, taking in the Acropolis Museum and doing a short stroll thru the Plaka neighborhood and stopping for a cool drink should be very doable, even in the heat.
