We have one night in Santorini, arriving at 7:45 am and leaving at 9:30 pm the next day. We have a wine tour starting at 3:00 pm and ends with a sunset viewing the first day. The following morning, we have a 5 hour catamaran tour that begins at 9:30 am. We would like to do a hike from Fira to Imirovigli, and we would like to see Pyrgos. Perhaps do the hike as soon as we arrive, and then see Pyrgos after the catamaran cruise? Will we have time, since we have to go through security at the airport for the 9:30 pm flight? This is leaving Oia out, unless we can do that after our hike. Any thoughts?
Well that plan seems to be overly enthusiastic. You are going to be on the move the whole time and you are not giving yourselves an opportunity to just be there, taking in the moment.
Having said that, yes you can do it the way you have suggested. First day do the hike. You probably are going to want to arrange a taxi in advance to get you back to your hotel from Imirovigli in time to refresh and get to the wine tour.
Since Imirovigli is at one end of the island and the wineries tends to be on the other side of the island you have to have your transfers nailed in advance.
However perhaps you can go to Pyrgos on day 1 wander around the village then take the wine tour as I suspect both are in the same vicinity.
Do the hike next morning after the catamaran tour which ends at 2:30. Do you know where the tour starts and stops. Options are the old port below Fira or at the ferry port. You will have to take where you end the boat trip in your plans. If you get off the boat at the old port you have to climb up the stairs about 900 meters or take the cable car. If there are cruise ships in port sometimes the line up can take an hour to get up waiting for the cable car. Hopefully you will have time to do the hike. You can probably arrange for your hotel to store your luggage so you can pick it up after you are finished. You should arrange for a taxi to pick you up so you can get your luggage and get to the airport for that 9:30 flight.
Good luck.
Do the Fira to Imerovigli hike on the morning of the first day.
It's a short hike (actually not really a hike, rather a long walk in the cobbled streets of the villages) and it will be better in the morning than under the afternoon sun, even in May.
Especially since the sun will still be (relatively) on the east side and you will be more in the shade than during the afternoon.
So yes, go to Pyrgos after the catamaran cruise on the second day.
Santorini is a small airport, the check-in and control time to be expected will depend on the number of flights departing that evening.
As a precaution, I would plan to arrive one hour before the flight, Especially since you probably won't have another flight to Athens after your 9.30pm flight in case you miss it.
If you are going to other islands like Naxos or Paros, or Sifnos or Amorgos, they have tons of walks & hiking trails... so you can skip that hike ... just do the walk from Fira to FIrostephani to enjoy the view. or from Oia to Imervoglii... no need to take hours and hours. THe reason that hike is promoted so much is because it's about the only one of note on this island. If Santorini is your only island, OK, do the walk. otherwise, don't crowd your schedule.