Let $A^*=Pic^0(A)$ and apply $Pic^0$ to a surjective homomorphism $J\to A^*$. Added: as noted, the existence of a surjective homomorphism goes back to a theorem of Matsusaka. It is proved in Milne, Jacobian Varieties, 1986, 10.1. As far as the duality is concerned, in terms of the functor $Pic^0$ it is rather obvious that the dual of a surjective homomorphism is injective. (If it is an isogeny, then one even knows that the kernel of the dual isogeny is the Cartier dual of the  kernel of the isogeny; see Mumford, Abelian Varieties, 1970, Theorem 1, p.143, where such things are explained in detail).