Answer
Yes, the demonstration would still work.
Work Step by Step
Yes, the demonstration would still work.
Let down be the positive direction. For a moment, let's suppose that the vertical acceleration is zero. Let the arrow's initial vertical velocity be $v_{0y}$ and let the arrow be a height $h_1$ above the monkey initially. Let's aim the arrow directly at the monkey and let $t$ be the time it takes for the arrow to hit the monkey. We can consider the vertical displacement of the arrow:
$\Delta y = v_{0y}~t = h_1$
With no vertical acceleration, the arrow would hit the monkey.
Now let's consider the same situation with a vertical acceleration of $g$. We can consider the motion of the monkey:
$\Delta y = \frac{1}{2}gt^2 = h_2$
We can consider the motion of the arrow:
$\Delta y = v_{0y}~t+\frac{1}{2}gt^2 = h_1+h_2$
In this case, the arrow would also hit the monkey.
