The Wellesley Inn has free broadband Internet access in all of its rooms. All you need is your laptop with built in wireless technology or a wireless card to plug into it.
The general aviation terminal also has free wireless internet access which one can access over in the Naples Air Center building. Again you need your own laptop etc etc...
The centre of Naples is rumoured to have free wireless internet access too... however I haven't tested it myself.
So there you go, just hope you have a laptop now!