Can my baby father get sole custody of our son even though he isn't on the birth certificate and not in his son's life?

You don't say how old your child is and whether the dad's failure to be in your child's life also includes not paying child support but regardless, unless you've abused or neglected your child the dad will not be able to take him away from you. If you concede paternity or he proves paternity he must pay child support. Unless there is a good reason not to, he should also get some type of visitation.