You would pay about the same as you do back home.
On all the cruises I have been on, if you give to your cabin attendent early in the day they usually have ready that evening.
Carnival Liberty New Year's Eve 2007
Liberty of the Seas 5/2008+11/2009
Allure 1/16/ 2011
Vision OTS 2-14-2015
Freedom OTS 3/6/16
We have been bringing DH suit in our carry on for the last few cruises, turn it in the first thing upon embarkation for dry cleaning and it is ready the night for formal night. It has been $8-$9. Much easier then trying to pack it and keep it nice.
This last cruise, I tried something I learned from this board. I cut a dry cleaning bag in fours and placed a sheet in between each item that would/could get wrinkled Everything came out beautiful, no wrinkles. Didn't have to use Downey Wrinkle Releaser once.
Teresa and Larry
#36-Carnival Splendor 9/16/12
7 night Mexican Riviera
#37-Allure of the Seas 11/11/12
7 night Eastern Caribbean