When do we have $A \subset B$ imples $f^{-1}(A) \subset f^{-1}(B)$? Im not sure if continuity of the function $f$ is enough to have the above. Or is it the monotonicity of $f$?

All you need is that $f$ is a function. If $x\in f^{-1}A$, then $f(x)\in A$, so $f(x)\in B$ as well.
Carl Mummert pointed out that if $R$ is any relation between $X$ and $Y$ then the claim still holds, where $R^{-1}A=\{x\in X: xRa\text{ for some }a\in A\}$. This is because if $A\subset B$, then $x\in R^{-1}A$ implies $xRa$ for some $a\in A$ implies $x\in R^{-1}B$.
