Many ports have internet cafes where you can purchase an hour of internet time for $5 or $6. Some you can find that have internet for the cost of a cup of coffee, or a beer. Just ask the crew, they usually know where the i-cafes are.
To use the ship's service you have to use one of their machines to swipe your Sea Pass, set up a wireless account and pick a user name and password to log in. Then once you open you laptop, select the ship's network, open Internet Explorer and a window will open to type in your username and password. Once that's done, you get a little popup that shows how much time you've used and I think how much you got left. I'd strongly recommend that you log in, download your email, log off (and I turn off the radio connection as well, I've read where some didn't log off correctly and received bills for $300 or more), while you're logged off, read your mail and write replies, then log on and send them, log back off again. Significantly cuts down on the time you're spending on the ship's network. Hope that helps.