Here's what I'm thinking...
A little neon tetra puts out so little waste into a 30 something gallon tank (sorry, us Americans are too lazy to learn the metric system), that the filter from the other tank should be able to handle it without a problem. I'd go ahead and do a 50% pwc just to lower the amount of ammonia that may be remaining from having the other fish in there. This way the tank will be temporarily livable for the neon for a while. You'll have some time to find another home for him without pulling your hair out.
The bad side to this is that while the bacteria in that filter will handle the single neon's waste, it's not going to be enough to feed the bacteria for very long. I'd try to find a new home for the neon within the next few days, and you've already got a fantastic head start on doing a fishless cycle since you have the seeding material.