I just figured it out! What I did was I created the batch file like I had it originally:
net use P: "\\server\foldername\foldername"
I then saved it to the desktop and right clicked the properties and checked run as administrator. I then copied the file to C:\Users\"TheUser"\AppData\Roaming\Microsoft\Windows\Start Menu\Programs\Startup
Where "TheUser" was the desired user I wanted to add it to.