Can we express the theory of a single topology as a multi-sorted theory?

Every topological space can be naturally construed as a model of your theory, but models of your theory aren't all topologies. For example, by downward Lowenheim-Skolem there is a model of your theory in which $X$ is infinite but $S$ is countable. Basically, you're only getting "definable approximations" to the powerset/iterated powerset.

This is the sense in which topology is not first-order: any attempt to "first-orderize" topology will result in things which look like topological spaces "up to first-order facts" but are not in fact topological spaces.

EDIT: That said, there are definitely ways we can bring first-order logic into play to analyze topological spaces. For example, we could look at first-order properties of associated structures such as the lattice of closed sets. See this article of Bankston for some results along these lines.