Take a clear coat nail polish and paint the ring, excluding the design. You just have to paint the band that goes around your finger. Once you coat it well, let it dry before wearing it back. The clear polish will prevent the metal and skin from keeping in contact and prevent discoloration.
It is always best to remove the source of the problem altogether. The discoloration happens when the metal reacts with moisture. First you have to prevent your fingers from getting moist while you still have the ring on. Apply an anti aspirant on your hands. Take the ring off your finger when you wash hands, take shower or do something that involves water. You also have to keep liquids like lotions away from your ring.
Soak your ring in a solution that is half ammonia and half water. Keep it there for two days. The ammonia will react with the copper and take it off the ring. This works perfectly for sterling silver rings with a little bit of copper.
You must have noticed that the green mark gets lighter as time passes. Most rings stop coloring altogether after some time. You just need to speed up the time till they do. One tip is to regularly wipe the ring with a clean tissue or towel. This way, the ring will stop coloring soon.
If you love your ring enough to take an extra step for it, go to a jeweler and ask him to coat your ring with Rhodium. Rhodium coat prevents the ring from tarnishing and ensures greater life with no green marks. Most fine jewelry is Rhodium plated.