Can I restore a nandroid backup that was made on an older OS? I was on Android 4.4.4 KitKat and I made a nandroid backup. Now I'm on 5.0 Lollipop and I'm wondering, do I have to reinstall everything and go through the setup  process again, or can I just use my 4.4.4 nandroid on my current 5.0 OS to get all my apps and stuff back?

I'm already rooted. I still have my recovery. I just need to know if I can restore an older backup to a newer OS or will it just revert me entirely back to 4.4.4.
[Can I just use my 4.4.4 Nandroid on my current 5.0 OS to get all my apps and stuff back?]

Take a look at Nandroid Browser:

With Nandroid Manager you can restore data from your nandroid such as apps+data, text messages, call logs, and much more.
...
Nandroid Manager currently supports nandroid backups created by TWRP and Clockworkmod (v5.xx and v6.xx) based recoveries.

If your backup meets the above criteria then go ahead (irrespective of 5.0 or 4.4) and use it to extract relevant data.

I just need to know if I can restore an older backup to a newer [OS] or will it just revert me entirely back to 4.4.4[?]

It depends upon what content you want to restore. If you choose to restore only Data then there could be some minor issues of app crashing or related stuff, but your OS won't revert back to 4.4.x. You still need to restore System, Boot, (possibly some other partitions too) to revert back to previous state.
Honestly, if the goal is to get the data, then I'll prefer Nandroid Browser. Else, I'll first take complete backup of my 5.0, then restore the Data of 4.4 on 5.0 to see whether everything runs smoothly.
Remember, do it in steps, if you choose to. It is possible that your Nandroid backup has some partitions whose size were changed during 5.0 update. Reverting haphazardly to 4.4 may lead to severe consequences. (e.g.: Yureka device hard-bricked when downgrading to Kitkat.)