I am running a basic micro instance on Amazon EC2. I am wondering if it is possible to run some remote GUI to work in the system as root?
If you're just looking for file system access you can use something like Midnight Commander which you should be able to install via yum or apt depending on which distro you're running.
If you're looking for more functionality - such as actually configuring the server, creating accounts, working with the firewall, etc, you could look at something like Webmin.
Lastly, if you want a full-blown remote desktop, you can install a desktop environment on your server (KDE, Gnome, etc) and then connect to it remotely using something like NoMachine.
I hope that one of those addresses your question.
When you say 'some remote GUI', what did you have in mind? You can use VNC,for example http://blog.gobansaor.com/2008/06/11/nx-rather-than-vnc-for-ec2-desktop/ or http://www.synergycode.com/services/oracle-on-amazon-ec2/item/set-up-vnc-and-remote-desktop-ec2