I wish that it were that easy today, but in the post 9/11 world, things have changed.
Any vessel taking departure or (especially) arrival in the US will be thoroughly vetted. This includes crew lists provided to Homeland Security (US Coast Guard) by the shipping company to allow for background checks and/or confirmation of identities of the signed on crew. It is not even possible in the US to gain access to the ship without proper documentation.
Sadly, in the 1st world in is no longer possible to simply show up at the head of the ships gangway and sign on as an ordinary seaman, wiper or food handler and work off the passage. In the developing world, well, who knows...It may be possible between some far-flung, off-the-beaten track ports, but I would not base travel plans upon that foundation.
In the US, an entry level seaman should plan on at least 6-12 weeks of leadtime, if applying for an original Merchant Mariners Document that would allow you to apply for work either through a national union or directly with the shipping company. In most cases, an entry level employee would not be granted privileges in terms of shipboard transportation of personal vehicles, senior officers are another story.
The days of casual labor in the maritime industry should be considered history.
I wish that it were different, but I would just work a little overtime in your current position and fly you and the bike to where you want to ride, it is afterall, all about the ride.