You can take your laptop and use it in your cabin if you desire. Check the details with Celebrity but I think I recall that using the connection in your cabin is around $ 3.00 to 3.75 a minute, whereas the internet cafe is a reasonable 75¢ per minute.
I believe you can compose your emails off-line and then go online to send them. We sailed on Celebrity's sister ship, the Millennium, and internet speed was excellent.
Have a wonderful cruise!