I visited Santorini about 7 years ago and when I took a semiprivate tour around the island. Next April I will be in Santorini again for 1 day coming On Celebrity cruise ship. We want to spend more time in Oia, go to the beach, maybe visit local winery (last time we visited Boutari). Maybe somebody made a similar trip and can advise - what beach to choose, winery to visit? How to travel from one point to another? Do island have any public transportation to go from Oia to the beach? What else to visit in Santorini? Thank you in advance.

I think you would get better responses over at Cruise Critic, ports of call. That being said, IMO you probably need to do Oia OR a beach. Both can be done by public transportation from a bus station in the main town but even getting to the bus station will take some time as you must get up there. You will probably have to tender off the ship and then take the cable car (or donkeys) up to Thira then find your way to the bus station - a 10 or 15 minute walk from where the cable car lets you off. Maybe if you rented a car / scooter / four wheel buggy you could fit in more but you'd be braver than me!

If you took a ship tour that started in Oia you might have time for both. We have stopped at Santorini on 3 cruises - one time we bussed to Oia, another to a beach (don't remember the name, red volcanic?) and one time we did a walk along the crater from Thira to Imeroglavi. You can walk all the way from Thira to Oia but that would be quite the day trip. But do plan to get out of Thira - total mob scene when big ship(s) in port!

I wouldn't waste any time on a beach there, nothing special as beaches go. If you get to Oia, you can walk( or taxi) down to Amoudi Harbor for a great seafood lunch.
Katina's is one I remember. Allow lots of time for those cruise ship rides on the gondola/ funicular. Often very long lines. I don't remeber the winery's name, but there was one near Oia. Buses run from Fira to Oia from bus station in Fira.

Do not even think of using the cruel and inhumane practice of donkeys to get to the top - that's Ehar the cable car is for - I am horrified post #1 even flagged it as an option��

Should say that's 'what' the cable car is for - there are plenty of informative posts on these forums alone about this cruel business - no one should ever use a donkey.

Sigalas Winery is in Oia.

Well said Blueeyedcod, heartbreaking to see those poor overworked horses carrying people up the steps.