Can anyone shed any light on how to set this function up in processing? I have seen some information about setting this up in Java using the Swing library. Is there any was of doing this in processing?

Perhaps look at the answers of the nearly duplicate question Text Help Required.

Processing authors strongly discourage the usage of Swing components in sketches, as it might mix badly with the PApplet code.
But you can do something similar with rect() and text() calls, with a bit of time management for set up (wait n milliseconds over target before displaying) and tear down (dismiss the tooltip after displaying it n milliseconds, or if going out of target).
Since the image is displayed in draw() with image() calls, you should draw the tooltip over it without modifying the image itself.