Since the menu items don’t get clicked on, they’re just text. The cursor is seperate. For the text, use JLabels in Swing or just draw it directly to the Graphics object.
You can make the menu JPanel a JDialog instead. If you make it modal, it should take and keep the focus, preventing the player’s keyboard input from getting captured by any other part of the interface. For drawing the text directly, you would extend the JDialog class and override the paintComponent method.
For changing the text, you can just change the text of the JLabels (I believe the correct method is called “setText”) or just change what text you’re drawing (if you’re drawing the text yourself).
You can use images if you want, but that’s not really any better than drawing the text directly.
If you like, you can avoid using Swing Components altogether and just draw the whole screen. You’ll need some kind of Frame (from Awt) or JFrame (from Swing), but you can draw everything youself.
I like using Swing just because it gives me a bunch of components to use.