Aromatic Apparent Substituion Reaction Mechanism The first molecule reacts with $\ce{NaOH}$ to form the second molecule. I am told that it does't undergo via a substitution mechanism. I am asked to draw the mechanism for it.

I am thinking that the double bond will attack the OH group (addition reaction) so that it will add onto the meta carbon.  Then the bromine group will leave (elimination reaction) which then the OH group attacks the para carbon. 

I am not really sure if that is correct. Could someone please help me out. Thanks.
The question is slightly misleading. It is usually referred to as a substitution; nucleophilic aromatic substitution to be precise; but it is actually an addition-elimination. What the question is trying to say is that it is not a regular $S_N1$/$S_N2$ substitution (which I think you correctly realised anyway).

The mechanism involves attack at the carbon attached to the bromine to form an intermediate anion which is stabilised by the electron withdrawing acetyl group, para to the site of attack.

I'll let you draw the arrows and come up with a complete mechanism.