OK, I think I figured this out thanks to the start I got from this post:
1. I added a function (code from the linked post above) to the functions.php file of my theme.
2. Then I copied the menu.png image from: wp-admin->images to my theme's image directory.
*you could probably link to this file directly without having to copy it but I haven't figured that out yet.
3. Then, in the CSS within the function that I added to my theme's function.php file, I entered the correct values to place the sprite for the icon that I wanted. The x value for each icon is 30 pixels different. So for the "Appearance" menu icon you set the x value to 0, the "Comments" icon is -30px, "Dashboard" icon -60px, etc. The x value is first and then the y value in the CSS declaration.
Passwords (maybe?): -360px
I hope this saves some people some time if they would like to set the menu icon of a registered CPT with one of the regular menu icons which come with WordPress.