You are concerned about not being passive-agressive, but to me it seems like your colleague is manipulating you. What exactly is this all about? He should be doing something as a part of his job... but somehow it became your problem, and you are asking on internet what you should do to solve it.
Of course people should help each other, but there are some limits where too much helpfulness opens door for exploitation. You can help him, but remember the limits; they could help you avoid some unpleasant surprises later.
First, don't let him shift responsibility to you. You can give him some good advice, for example teach him to use a reminder system. It's his job, he is paid for doing it, and he is paid for doing it correctly. -- If your work will require you to do something once in a month, who will be responsible for remembering it correctly? You will.
I don't know what "task" he is doing monthly and why exactly some "specific systems" should not be included. But I assume that this request is not your personal whim, but is somehow necessary for a work that your employer wants to be done. You should document this necessity in some report, so your manager knows about it. Without mentioning your colleague. Just write: "To make X work correctly, it is essential that Y is not done on server Z; otherwise it might stop working."
If you accept that sending reminders is your responsibility, suddenly every possible problem related to the reminders becomes your fault. What if your Outlook for some reason does not send the reminder? (There could be a bug, a network problem, an update that does not preserve configuration, etc.) Or what if your colleague simply claims that he did not receive the reminder, without any evidence either way? By accepting the responsibility for the reminders, you invite all these problems on yourself.
Second, be careful about subtle power games in the workplace. Just because you avoid them, it does not guarantee they will avoid you. Accepting a task from your colleague is a very bad move. Think about it this way: "If company thinks about promoting one of two employees A and B, and A is already giving tasks to B, which one of them is more likely to be promoted?" If you accept a subordinate position, it will be only natural for everyone else to confirm it.
But for the same reason your colleague should not accept tasks from you. Which is why this situation should not be framed as something between you and him, which you two have to negotiate on. -- You did your work. Your work needs your systems to be excluded from the monthly task. You should explain this need to your manager. Your manager should ask your colleague to exclude the systems from the monthly task. -- This is the proper way to do it. Then it's between the manager and your colleague, and you are out. And I don't think your colleague would ask his manager to send him reminders.
So at this moment you should 1) explain to your manager that your systems need to be excluded from the monthly tasks, preferably in a written way; 2) tell your colleague that you will not send him reminders, because you have a lot of other work to do, and 3) offer your colleague to help him setting up his reminders. Whether he accepts or refuses your offer, it's no longer your problem.