I think that this can be extracted from the first chapter of Beauville's book on algebraic surfaces. First note that if D and D' are smooth irreducible curves that meet transversally, then the identity $C(D) \cup C(D') = D \cdot D'$ holds and is equal to the number of set-theoretic intersection points of $D$ and $D'$. This is one of the standard ways to compute the intersection product in smooth topology, recall that transversal intersections of complex submanifolds always contribute $+1$. The right hand side is equal to this by Definition 1.3 of the intersection form (which is proven to be equal to the sheaf-theoretic definition in Theorem 1.4).

The issue of finding transversal representatives is treated by Fact 1.7 (which is originally a theorem of Serre). It states that any divisor $D \subset S$ may be written (up tp linear equivelance) as $D= A-B$ where $A$ and $B$ are smooth curves, $A$ is a hyperplane section and $B$ is a positive multiple of a hyperplane section (for a possibly different projective embedding). The advantage of having hyperplane sections is that we can peturb them (by perturbing the corresponding hyperplane). This gives us enough freedom to find transversal representatives for any two divisors (up to a linearly equivalent, possibly non-effective representatives), and allows us to reduce to the above case.

I think that this can be extracted from the first chapter of Beauville's book on algebraic surfaces. First note that if D and D' are smooth irreducible curves that meet transversally, then the identity $C(D) \cup C(D') = D \cdot D'$ holds and is equal to the number of set-theoretic intersection points of $D$ and $D'$. This is one of the standard ways to compute the intersection product in smooth topology, recall that transversal intersection points of complex submanifolds always contribute $+1$. The right hand side is equal to this by Definition 1.3 of the intersection form (which is proven to be equal to the sheaf-theoretic definition in Theorem 1.4).

The issue of finding transversal representatives is treated by Fact 1.7 (which is originally a theorem of Serre). It states that any divisor $D \subset S$ may be written (up tp linear equivelance) as $D= A-B$ where $A$ and $B$ are smooth curves, $A$ is a hyperplane section and $B$ is a positive multiple of a hyperplane section (for a possibly different projective embedding). The advantage of having hyperplane sections is that we can peturb them (by perturbing the corresponding hyperplane). This gives us enough freedom to find transversal representatives for any two divisors (up to a linearly equivalent, possibly non-effective representatives), and allows us to reduce to the above case.