So, I've been trying to figure out how to get the same issue (well, not really an issue, just a pet peeve of mine) resolved.
On my Note 2, I was able to only get the phone to put a notification in the notification bar and not have a popup or wake the phone. For whatever reason, Samsung changed around the software, so that no matter what I do on my Note 3, the reminder will wake the phone up and show the event on the screen.
For me personally, I find any wake function like this to be completely annoying, especially since I keep my phone in my pocket. It's the same reason I had searched for an app to prevent text messages from waking up the phone (I use SMS Pop-up to prevent the phone from waking up).
So, I did some digging and found this app:
That app allows you to set the notification so that it doesn't wake up the phone! I've set it up, so that I don't get a pop-up either. I just have a notification in the notification bar, the notification sound, and the led alert. That's all I needed. For now, this app has served what I was trying to do, but if anybody knows how to prevent the phone from waking up (either S4 or Note 3), please chime in. I prefer not having apps that only serve single purposes, but in this case, this app was very important for me.