Doesn't look like there is enough time for your BF to get his passport unless he pays for expedited service which is pretty expensive. You probably wouldn't have any problem finding someone to go with you since it's free.
Don't you have some girlfriends or relatives that would like to go to Jamaica with you?
Of course you could just book it for two people and then just go yourself and say the other person couldn't make it. But what a waste.