I think you won't be able to stop access to Advanced Startup for one user, because one way to get there is from the Login screen itself, before you've actually logged in as any user (whether Administrator or Standard).
(You right click on the "power button" icon that's in the bottom-right corner of the login screen, and hold the Shift key down as you click on Restart.)
Once in Advanced Startup, some options seem to require Administrator access, but when I tried it just now, I could get a long way into Reset before I chickened out (the screen where you choose whether you just want to wipe the Windows partition) without it asking me for a password.
I certainly got further than Step 3 in this Tutorial, which is when I expected to be asked for an Admin-level password. (My PC has both Admin- and Standard-level accounts, all with passwords.)
http://www.eightforums.com/tutorials/2302-reset-windows-8-a.html
However I was doing this with a Windows Setup DVD in the CD drive; it may be different if the user is accessing files on the local PC's Recovery Partition instead and doesn't have access to a Setup DVD. But it's not hard to get hold of the images to make a Windows Setup DVD - for one thing the Windows 8.1 Preview currently has ISO versions which you can download freely, which I strongly suspect would have the capability to Reset your computer and lose your information.