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Harald, as Peter May observes in his notes on finite topological spaces (but as must already be standard), your partially-ordered pre-ordered sets provide the only examples. Indeed, if $X$ is an Alexandrov space (or Alexandrov-discrete space, whatever the terminology is), then we may impose a partial order pre-order on it by demanding for $x, y \in X$ that $x \le y$ if and only if $x$ lies in every neighbourhood of $y$. This becomes a genuine partial order exactly when $X$ is $T_0$.
Harald, as Peter May observes in his notes on finite topological spaces (but as must already be standard), your partially-ordered sets provide the only examples. Indeed, if $X$ is an Alexandrov space (or Alexandrov-discrete space, whatever the terminology is), then we may impose a partial order on it by demanding for $x, y \in X$ that $x \le y$ if and only if $x$ lies in every neighbourhood of $y$.