You cannot go from a newer version to an older one (or it is extremely difficult), so it is better to backup your personal data and reinstall instead of trying that.
I would also say that Ubuntu 12.04 LTS is well polished and debugged, and if you do not need drivers for very new hardware, it is the best option for a production platform (a computer that should work in a reliable way).
But there are also other GUIs or desktop environments, that might serve you well in 13.04. You can install them and select the one you like at the log in screen (click the round symbol near where you enter the password if you have standard Ubuntu).
You can also download the corresponding iso file, make a boot CD/DVD/USB drive and try it live without installing anything.
sudo apt-get install lubuntu-desktop
sudo apt-get install xubuntu-desktop
sudo apt-get install kubuntu-desktop