Short answer: Maybe, if you pay extra
Your phone should be in airplane mode if you do not want a huge data bill.
Internet is considered a luxury on a cruise. Fast speeds are just starting to catch up with cruise ships. There are computer cafes available on the ship, but you do need an internet package to use those as well.
Each cruise line has different internet packages. Carnival has 3 from just social media package to a streaming package. Royal Caribbean just introduced VOOM. You get a discount if you buy the internet package before getting on the cruise.
You can buy it on ship as well. However, you cannot buy for one day unless you purchase it on the last day of the cruise.