Use of the cruise ship's computers is free. It's internet access for which you have to pay, whether you use a ship's computer or your own. Typically, ships offer internet options, such that the more time you buy, the lower the per minute cost. But everyone pays the same for using the internet, whether or not you use your own computer.
I've never sailed Ryndam so I can't speak about where the hot spots on that ship are located.