"If I get the dual citizenship then because he is married to me can that pass on to him as well?"
I'm not 100% sure how this works in Jamaica but in my case I can have dual citizenship/a Euro passport but my husband cannot. If I chose to live in the Union I would imagine I could get some residency status for him based on our marriage but unless he applied for citizenship he wouldn't automatically get that being married to me.
I suggest you make phone calls. Jamaica is a bureaucrats dream and getting the same answer to one question from multiple people can be a challenge. Maybe work in some appointments/visits to the proper offices when you go down to start checking out rental properties to get the process going and get some of your questions answered.