If there is adequate, in fact a lot of rain in a year, is it natural (colloquially) to say:
It rained well this year.
It rained good this year.
Do the bold words sound natural?
Neither of those sound particularly natural. The second is certainly wrong, whereas the first sounds more as if you are discussing how the rain fell, rather than how much of it there was. Colloquially, you're probably just say "It rained a lot this year," or perhaps "It was a very wet year."
In the US, you would actually say something like "we had a normal amount of rain this year" or "we had an average rainfall this year". This is, of course, compared to the historical rain totals for a location. New Yorkers would have a different sense of normal compared to Los Angeles residents.