Is my client truly in danger from her black upstairs neighbor who has called her a racist and has threatened to have a friend kill her? What is her best course of action?

You saw intuitively that this is simply bullying and a baseless threat for the moment. That is not to say he will never act on this impulse, but he is not at that point where he’s willing to risk himself, even indirectly, in a homicide within his building. That is not to say this should be taken lightly, but your suggestion to her to not inflame the situation by engaging with him and leading to a contradiction let alone a confrontation, is the wisest course—just steer clear and do nothing to provoke him. We are working to contain things and that can work, barring a misstep of some kind or a worsening of his mental status.