That's a nice insight over the GUI internals. Unfortunately, that logic will become very complex as we add more dropdowns to the same GUI - for instance, if you are creating such buttons in a loop.

A shorter workaround is to give up using GenericMenu.DropDown and use GenericMenu.ShowAsContext instead. If you have a dedicated UX team, they will give you a long speech about using the proper UI elements and alignments. If you are just creating something to improve your team's productivity, nobody will notice the difference...

This is because of when the DropdownButton returns true, and when you're allowed to call GUILayoutUtility.GetLastRect.

You can only call GUILayoutUtility.GetLastRect during a repaint event (Event.current.type == EventType.Repaint), otherwise it returns dummy values. (This is what you're experiencing.) The DropdownButton also only returns 'true' on the "used" event (Event.current.type == EventType.used). Because of both of these, you can't have the GetLastRect call within the if statement for the DropdownButton. This won't work:

if (EditorGUILayout.DropdownButton(new GUIContent("Click me!"), FocusType.Keyboard)) {
// GetLastRect only returns real values during Repaint
if (Event.current.type == EventType.Repaint) {
// This code is never reached, because DropdownButton only returns true on Used
Rect menuRect = GUILayoutUtility.GetLastRect();
Debug.Log("I never actually get called!" + Environment.NewLine + menuRect.ToString());
}
}

But by splitting out the GetLastRect so that it occurs regardless of the DropdownButton being pressed, it will indeed work.

private Rect menuRect;

public override void OnInspectorGUI() {

bool selected = EditorGUILayout.DropdownButton(new GUIContent("Click me!"), FocusType.Keyboard);

if (Event.current.type == EventType.Repaint) {
menuRect = GUILayoutUtility.GetLastRect();
Debug.Log("I get called now!" + Environment.NewLine + menuRect.ToString());
}

if (selected) {
GenericMenu menu = new GenericMenu();
menu.AddItem(new GUIContent("I'm drawing in the correct position!"), false, null);
menu.DropDown(menuRect);
}
}

Hopefully this helps anyone else stumbling upon this problem.