Counter-example: If $J$ is prime then $f^{-1} (J) $ is prime. $f$ need not be unital. Let $R$ and $S$ be commutative with 1 and $ $ $ f:R\rightarrow S$ is a ring homomorphism which need not be surjective or unital i.e. $f(1_R)=1_S$
I know that for surjective or unital ring homomorphisms the statement - "If $J$ is prime then $f^{-1} (J)$ is prime" - holds true. 
However, if f need not be surjective or unital, are there any counter examples for the above statement?

Yes: the zero map would then be a morphism of rings, and the preimage of every prime ideal is then $R$, which is not a prime ideal (by definition)
