Civil registration of births in Ireland started in 1864 so there will be no birth certificate for James Nelson born c1841. The normal procedure is to search for church records of a baptism but even if you did know the family's religion there are numerous possible churches in Belfast (not sure if all of them would have records for that period and baptismal registers for that period sometimes list only minimal details like date of baptism, name of child, names of father so at the moment I don't think church records is a practical option).
One thing you can try is old Belfast directories (see
www.proni.gov.uk) as father James might be listed- this is assuming he was the proprietor of a pub rather than merely working in a pub.