I tried to solve 1.3.3 in Bosch, Algebraic Geometry and Commutative Algebra.
I did not find a way to solve it. But I found this: Finitely many prime ideals ⇒ cartesian product of local rings.
And I am not able to show that in a commutative ring with unit, $R$, which has only finitely many prime ideals and nilpotent nilradical the Jacobson radical is also nilpotent.
I would be happy if someone could give me a hint how to solve it. Thanks. You can find the exercise from Bosch in the link.