I wanted to post a comment to Tom's answer, but couldn't figure out how to do it, so I add it as an answer...
I think you can interpret the classical Nullstellensatz in this language as follows: Take $\mathbf{Sp}$ to be the opposite of the category of reduced finitely generated $k$-algebras, and extend it to have arbitrary co-products (so a full sub-category of affine $k$-schemes). Take $\pi_0$ to assign to each algebra the set of maximal ideals, $D(A)=\coprod_Aspec(k)$, $U(X)$ the set of $k$-points. The map from $U$ to $\pi_0$ assigns to a $k$-point the corresponding kernel. Then the usual Nullstellensatz says that this map is surjective if $k$ is algebraically closed.
I didn't think what $I$ should be in this case...
