Covering maps of Riemann surfaces vs covering maps of $k$-algebraic curves

No, the property of having small neighbourhoods whose preimage is a disjoint union
of $n$ homeomorphic open sets does not hold in the Zariski topology (once $n > 1$, i.e.
the cover is non-trivial). The reason is
that non-empty Zariski open sets are always very big; in the case of a curve,
their complement is always just a finite number of points. In particular, two different
non-empty Zariski opens are never disjoint.

There are two ways that one rescues the situation: the first is to use the differential
topology view-point on covers: they are proper submersions between manifolds of the same dimension.
In the context of compact Riemann surfaces, both source and target have the same dimension,
and maps are automatically proper, so it is just the submersion property that is left
to think about. It is a property about how tangent spaces map, which can be translated into
the algebraic context (e.g. using the notion of Zariski tangent spaces).

So if $f: X rightarrow Y$ is a regular morphism (regular morphism is the algebraic geometry terminology for an everywhere defined map given locally by rational functions) of projective curves, we can say that
$f$ is unramified at a point $p in X$ if $f$ induces an isomorphism from the Zariski tangent
space of $X$ at $p$ to the Zariksi tangent space to $Y$ at $f(P)$.

For historical reasons, if $f$ is unramified at every point in its domain, we say that
$f$ is etale (rather than a cover), but this corresponds precisely to the notion of a covering map when we pass to Riemann surfaces.

This leads to the more sophisticated rescue: one considers all the etale maps from (not necessarily projective or connected) curves $X$ to $Y$, and considers them as forming a topology on $Y$, the so-called etale topology of $Y$. This leads to many important notions
and results, since it allows one to transport many topological notions (in particular, fundamental groups and cohomology) to the algebraic context.