Given a topological space $(X,T)$, the topology $T$ is also a partial order with the inclusion relation $(T,\subseteq)$.
Given a continuous function $f:A\to B$ between two spaces $(A,T_1)$ and $(B, T_2)$, where $T_1$ and $T_2$ are topologies. $f$ induces a map $f_*:T_2\to T_1$ where $f_*(S)= f^{-1}(S)\in T_1$ for each $S \in T_2$.
Is $f_*$ always order preserving?
If $f$ is an open map, we also have the map $\bar{f}:T_1\to T_2$ such that $\bar{f}(S)=f(S)$ for $S\in T_1$; is $\bar{f}$ order preserving?
As a side question (Edit: now, the main question), is this way of thinking about topologies as partial orders "useful"? Is there a nice result using the intuition about thinking topologies as partial orders?
Yes, $f_*$ and $\bar{f}$ are both order preserving, but this has nothing to do with topology. Given any function $f \colon X \to Y$ between sets $X$ and $Y$, then $f_* \colon P(Y) \to P(X)$ and $\bar{f} \colon P(X) \to P(Y)$ are both order preserving. Here $P(X)$ denotes the power set of $X$.