Assuming that the branched locus of the map $C\to \mathbb{P}^1$ contains only points $b\in \mathbb{P}^1$ corresponding to smooth fibers of the map $S\to \mathbb{P}^1$, the surface $S_1$ will be a non-singular elliptic surface fibered over $ C\cong \mathbb{P}^1$ and will have Euler characteristic exactly $24d$ (another way to say this condition is that the map $C\to \mathbb{P}^1$ is transverse to the map $S\to \mathbb{P}^1$). If the branched locus includes points corresponding to singular fibers, the surface $S_1$ will be singular. The easiest method to compute the Euler characteristic of $S_1$ (in both the singular and the non-singular cases) is motivically: Euler characteristic is additive under stratifications and multiplicative for smooth fiber bundles. So you stratify $S_1$ by the topological type of the fibers of $S_1\to C$. The open strata has euler characteristic zero since the fibers are elliptic curves and so the euler characteristic of $S_1$ is the sum of the Euler characteristics of the singular fibers. In the case where the map $C\to \mathbb{P}^1$ is transverse to $S\to \mathbb{P}^1$, the singular fibers of $S_1\to C$ are just $d$ copies of the singular fibers of $S$ and hence $e(S_1)=d\cdot e(S)$. In the singular case, some of the singular fibers may have multiplicity and hence there may be fewer than $d$ copies of a singular fiber of $S$. Thus in the singular case, $e(S_1)<24d$.
By the way, the condition that $C$ is contained in $S$ is a bit of a red herring here. The surface $S_1$ only depends on the maps $C\to \mathbb{P}^1$ and $S \to \mathbb{P}^1$, and not at all the embedding $C \subset S$.
