They do not provide corkscrews in the cabins. However, you can take it to any of the bars and ask a bartender to open it for you, providing you let them know it is for your cabin.
I've only had one bartender to tell me there would be a corkage fee. This was at the Lobby Bar on the Fascination. I nicely straightened him out (he must have been new) regarding the policy about corkage fees.
The cabin steward brought me one once when I asked. All I know is I asked and he started yelling at someone down the hall and then he brought it and I gave it back to him.
I don't see corkscrews on the prohibited items list as far as the ship. So you could probably just take a cheap one with you. Not sure about the Airline regulations. I've taken one on board without any problems.
I noticed Hookah's are now on the list. This guy will be disappointed.
I keep one in my luggage at all times so I don't forget it. You wouldn't believe how many times we have used it when there is none available. I've made a few people happy over the years. No problems in the air (checked) or getting on the ship.
Curiously, I just read a report where a guy said TSA was fine with his cork screw, but when he got to Heathrow and had to go through security again, UK security confiscated it as it is against their rules.
So if you are flying to Europe, I'd put the cork screw in checked luggage.