To begin, remove the strap and use a wide-bladed case knife to remove the snapback. I have the best results by rotating the blade up from the groove, pivoting off the back of the knife on the lugs.
This is what the plastic spacer ring looks like.
Use a plexi stick or similar to pry the ring up and away from the movement and case, until it's sitting free on top of both.
Any tweezer will do to remove it, though I'm particularly fond of bronze for its grip and gentleness.
Here's the watch with the ring removed.
Use a very soft tweezer (plastic or wood) to put the new ring in place, or covered fingers (finger cots/rubber gloves). The raw, highly-finished brass is incredibly delicate, so it scratches instantly and will immediately tarnish if touched with bare hands... But it glitters brightly when kept in good condition!
The ring is a close fit to both the movement and the case, so it needs to be pressed into place. A soft plexi stick or covered fingers will work wonderfully. It should seat fully into the case, and lie just below the plane of the movement.
One of the trickiest bits will likely be closing the case up again once the ring is installed—I've always had to use a caseback press to get it snapped on, but firm pressure with your thumbs on the back (with the crystal on a casing pad) will likely do if you press hard enough!