(a) The inequality \(|z - 3 - 2i| \leq 1\) represents a circle on the Argand diagram with center at \((3, 2)\) and radius 1. The inequality \(\text{Im} \, z \geq 2\) represents the region above the line \(y = 2\). The shaded region is the intersection of the circle and the half-plane above \(y = 2\).

(b) To find the greatest value of \(\arg z\), consider the point on the circle where the argument is maximized. This occurs at the point where the tangent to the circle is vertical. The point on the circle with the greatest argument is \((3, 3)\). The argument is given by \(\arg z = \arctan \left( \frac{2}{3} \right) + \sin^{-1} \left( \frac{1}{\sqrt{13}} \right)\). Calculating this gives \(\arg z \approx 49.8^{\circ}\).