Well basically, you have w = (z+2i)/(2iz-1)
And so arg(w) = arg(z+2i) - arg(2iz-1)
Then you plot the vectors z+2i and 2iz-1. Notice that it is implied that w must be parallel to z+2i rotated clockwise by arg(2iz-1).
Now, as z rotates anticlockwise from -i to i, we have arg(z+2i) moving from pi/2, increasing as z approaches 1 then decreasing back to pi/2 as z reaches i. At the same time, arg(2iz-1) increases from 0 to pi. Therefore, we have arg(w)=arg(z+2i)-arg(2iz-1) decreasing from pi/2 to -pi/2, clockwise. Thus as z moves anticlockwise from -i to i, w moves from i to -i -- the opposite direction. Using the same method, it is shown that when z moves from i to -i, w moves from -i to i, i.e. the opposite direction.
Thus, we have shown that as z traces the unit circle in one direction, w traces the unit circle in the other.
(Note: The unit circle part you have already proven. Just express z in terms of w, then take the modulus of both sides, equating it to 1.)