Assume that we have a Galois connection formed by two monotone maps $f\colon X\to Y$ and $g\colon Y\to X$.
I want to know whether the following statement is true: if $f$ is bijective, then $f$ is an order-isomorphism, that is, $f(x)\leq f(y)$ implies $x\leq y$. I don't find any example in which it fails.
Since you mention monotone maps forming a Galois connection, I assume that you mean
$$f(x) \leq y \Leftrightarrow x \leq g(y).$$
(In this case, we say $f$ is the lower adjoint of the Galois connection and $g$ is the upper adjoint.)
Among other consequences, $fgf=f$ and $gfg=g$.
Now from $f(x) \leq f(x)$ follows that $x \leq gf(x)$; but if $x < gf(x)$, then $f(x) < fgf(x) = f(x)$, because $f$ is injective. Hence $x = gf(x)$.
So if $f(x_1) \leq f(x_2)$, then $x_1 = gf(x_1) \leq gf(x_2) = x_2$, as you wanted.
