Mobile phone signals work on frequencies (and at power levels) which only carry the signal very short distances. Additionally, radio waves on these frequencies won't carry beyond the horizon. So,you can't use a mobile phone while at sea (unless, of course, you're very close to the coast).
The only way you could get an internet connection out at sea would be via a satellite link. Satellite phones are very expensive and normally require that a portable satellite dish is positioned on stable ground (which you obviously don't have on board a ship). So, even if you'd got the mega-bucks to buy a satellite phone, it still wouldn't help.
The only possible route would be a link via the ship's satellite communications system. This would be similar to the systems used by transatlantic sailors to send e-mails to their families. Such data links are very slow, with a simple e-mail of just a few lines often taking several minutes to send (so web-browsing is effectively ruled out).
Chris