I'm not sure the internet is working at Pedro's currently. It certainly isn't on the desktop machine that's used for playing music.

Pedro's is an evening bar. Limes is an all-day bar, and as Tacoboy says above has both wifi and mains connections. They also have a computer for customers' use. If you don't know Limes, it's directly opposite the new Tropic Air terminal, directly north of the entrance to Sunbreeze Hotel.