Due to the fact that Cuba is a communist country and the dictator in Cuba won't let any of the important genealogy information that the Churches and civil registration in Cuba has to be online, it's very difficult to obtain information on your ancestor. Below is a link to a company that can get you any type of old papers from Cuba, but it's not cheap.
http://cubacityhall.com/According to this Cuban genealogy website there's quite a bunch of Fuller in Cuba.
http://www.cubagenweb.org/ships/php/pass_search.php?in_surna...Good Luck!