As the question states, I have to show that the category Ord is not balanced.
Doing so involves constructing a morphism $f$ which is monic and epic but is not an isomorphism. In Ord, the functions which are monic and epic are exactly the functions which are both injective and surjective. So consider $A = \{ a, b \}$ with $a \leq b$ and $b \leq a$ and $B = \{ a, b \}$ with $a \leq b$. Now consider the monic and epic morphism $f : A \to B$ defined by $f(a_A) = a_B$ and $f(b_A) = b_B$. I claim that there is no inverse to $f$ which preserves the ordering. Does this make sense?
I suppose you probably wanted to go the other way around, otherwise the $f$ is not preorder preserving: you have $b_A \leq a_A$ but $$f(b_A)=b_B \not\leq a_B=f(a_A)\ .$$
Nevertheless is you reverse the roles of $A$ and $B$ meaning that you let $$\begin{align*} f \colon B &\longrightarrow A\\ f(b_B) &= b_A \\ f(a_B) &= a_A \end{align*}$$ the you actually get bijective preorder-preserving map, that does not have an inverse preorder preserving (I assume that the details are obvious to complete).