Unanswered questions are simply those that have no accepted answer and have no answers with positive vote counts. It isn't necessarily the fault of the asker for not accepting an answer. If a question with answers shows up on the "unanswered" page, then that means not only that the asker didn't accept one, but that nobody else who saw the answers thought they were worthy of votes, either.
There are other ways to reduce the number of unanswered questions that don't involve automatically punishing or nagging the people who asked them:
As you've already done, answer them. If anybody sees your answer and votes it up, then the question will cease to be unanswered and you will have accomplished your goal. It doesn't require anything from the asker to make that happen.
Find "unanswered" questions that have answers, find an answer that you think is good, and vote it up. That will give it a positive vote count, and the question will no longer be considered unanswered.
(Maybe those are what you meant by "do something with old questions," but that's not the impression I got.)
I said above automatically punishing or nagging. Askers are already punished and nagged manually by people who notice low accept rates. I don't think the process should become any more formalized than that.
I've already seen cases where people have accepted an answer that didn't fully answer the question, sometimes even one that didn't work at all. Sometimes, these people did it because they felt pressured to accept them for fear of their accept rates to appearing too low. I fear that the suggestions given in this question would just make that situation worse.