Does the Dunning-Kruger effect still work the same if the incompetent person is aware of this effect? Is it possible for an incompetent person to reevaluate his abilities (to become more realistic about his abilities) when finding about the Dunning-Kruger effect?
One critique of the Dunning-Kruger effect is that it is just regression to the mean. I.e., there is an imperfect correlation between perceived an actual ability. A consequence of this is that people low in ability will overestimate and people high in ability will underestimate. Of course, in some domains, we also see a leniency bias whereby people in general have an inflated sense of their ability. For a discussion of these interpretations see this post by Tal Yarkoni.

But in general, there are ways of increasing the correlation between perceived an actual ability. The basic principle is to provide people with feedback on their performance that is accurate. And presumably, it would help to persuade people that it is trustworthy.