Do not attempt to escalate passive-aggressive behavior. The other person is more skilled at it, and you will lose. There is really no safe way to respond to passive-aggressive notes.
Respond factually and do not allow the other person to push your buttons.
Have a face-to-face conversation without assigning blame. Blame will provoke excuses. Simply pointing out the other person's passive-aggressive behavior is not enough; they will deny that they are behaving this way, no matter how much evidence you may present them with.
Set clear expectations, assign consequences for not completing tasks, and be sure to get everything agreed to in writing. Keep signed copies, as with a contract. That way the other person will have no option but to work with you, and if they forget, you can refer to the signed agreement. (Of course, the other person will have plenty of excuses why they could not fulfill the agreement, but then you have the consequences to rely upon.)
If you can sense the other person's frustration or anger, you may have some success in empathizing with them. Saying something like, "I know it must be frustrating dealing with _________" may help, by recognizing the other person's suppressed feelings.
Take choice away from them. Say, "If I do not see X result by this date, this will happen." Do not attempt to argue with them, just lay down the decision.
Always have a backup plan in case of procrastination.
Do not accept bad behavior, but call it out publicly (in a nice way). When the sighs and eye-rolling start, ask, "Do you have something you wish to contribute?" or something similar that is appropriate.
If possible, enlist a neutral third party who sees the problem firsthand. Sometimes the weight of public opinion will help sway a person's passive-aggressive behavior.