If, as I believe, Sun Princess has the same internet setup as her sister, Dawn Princess, you will be charged $7.50 for up to 15 minutes usage. So anytime online from 1 minute to 15 will cost $7.50. Sixteen minutes usage would cost you $15. Unfortunately, access to computers, unless you choose to rent a laptop at the purser's desk (and I think you are restricted to using it in closer proximity to that area) is available only in the business center which has three or four computers which are available only at limited times during the day. We frequently had to wait in line for lengthy periods before we could use them. I believe that the larger Princess ships have internet cafes with a larger number of computers available 24/7. I was much happier on Royal Caribbean's ships which have many workstations available 24/7, charged $0.50 per minute and billed you on a per minute basis rather than in 15 minute increments.
If you are a Platinum member of the Captians Circle you can get free internet access on a space available basis. Based on my past experience with Princess (eighth cruise with Princess comming up) there are lots of off peak hours that nobody is using the laptops they provide at the pursers desk.
It is also possible for your friends to send you email for free. They will need to know what ship you are on and your cabin number. It needs to addressed to the ship and your name and cabin number need to appear in the subject line, for example: