In the first sentence, exactly modifies the verb addresses, and in the second sentence, exactly modifies your question. These mean nearly exactly the same thing.
this paper exactly addresses one of your questions,
this paper addresses exactly one of your questions.
In the first one, exactly modifies addresses, essentially meaning that the paper contains an analysis of one of your questions, without having to do extra work to fit the question to the paper. But it might say something about some of your other questions.
In the second one, exactly modifies one, so the paper doesn't address two of your questions, or none of your questions.