Step 1: Draw the line of sight.
The observer stands at height $80$ m. The angle of depression to an object equals the angle of elevation from that object back to the eye, so we can use ordinary right triangles.
Step 2: Use the bottom of the flag first.
The bottom of the flag sits on the ground. Its depression angle is $45^\circ$, so $\tan 45^\circ=\frac{80}{d}$ where $d$ is the horizontal distance. Since $\tan 45^\circ=1$, we get $d=80$ m.
Step 3: Set up for the top of the flag.
Let the flag have height $h$. The top is $80-h$ below the eye, at the same horizontal distance $80$.
Step 4: Use the top's angle.
The depression to the top is $30^\circ$, so $\tan 30^\circ=\frac{80-h}{80}$, and $\tan 30^\circ=\frac{1}{\sqrt 3}$.
Step 5: Solve for the vertical drop.
$80-h=\frac{80}{\sqrt 3}$.
Step 6: Isolate the flag height.
$h=80-\frac{80}{\sqrt 3}=80\left(1-\frac{1}{\sqrt 3}\right)$, matching option 3.
\[ \boxed{80\left(1-\frac{1}{\sqrt 3}\right)} \]