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Which is correct:
1-He talked about the problem not unwillingly.
2-He talked about the problem not very willingly.
3-He talked about the problem not willingly.
4-Not willingly, he talked about the problem.
5-He talked about the problem, not willingly.
I would only use #1. Even though it is a bit clumsy, it is an example of "litotes", negating a negative to make a weak positive. The first is less strong than "He talked about the problem willingly."
I think reluctantly works much better than not unwillingly. That is because that is what the writer is saying only it says it more directly. While I understand what litotes is, I don't think the example sentence is a very good example of it.
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