You will have to use the terminal and follow any installation instructions
--you can make a comparison to the other Linux versions mentioned, even if they don't exactly match, the aTunes start command and the general method will be valid (whether it is an .rpm or a .deb) file
If you decide to try the .deb file from that aTunes download page..
You can in a terminal (provided the supporting applications are already installed)
--you will already have jre (java runtime envirnoment)
In terminal, check
java -version http://www.atunes.org/wiki/index.php?title=FAQ
--check if java is installed and its version
sudo dpkg -i packagename.deb
--the downloaded package (name) from aTunes
Linux Mint 7(gloria) and 8(Helena) main edition is a Gnome based desktop, based from Ubuntu 9.04(jaunty) or 9.10 (Karmic Koala) You might want to get the latest stable version, as released..
--you could try the .deb package, making sure that you have already installed the dependencies it(aTunes) requires
Java 6 Runtime Edition (http://www.java.com/download/)
For Linux and Mac users: mplayer or xine, cdda2wav. For ripping, lame, flac or oggenc must be installed.
--the only items/applications you might not have already installed in Linux mint 7 or 8 is For rippingflac or oggenc must be installed.