I've used a wire wheel on a slow speed Dremel before to clean up surface corrosion on contacts like that, just take it slow and gentle and make sure the brush is spinning clockwise so that it doesn't try to pull the metal away from the plastic housing.
I would try the other methods mentioned first though!
Front loading NES contacts are notorious for being flaky, if cleaning does not help, at least you can buy replacement cart slots. you would need to de-solder it and solder the new one. and I wouldn't recommend it for a beginner. lots of traces to break!