As stated above Kstars runs under the KDE Desktop Environment. In fact it is part of the standard set of applications that is shipped with KDE (well the kde-edu part anyway).
KDE will run under various unices, including Linux, BSD, Solaris (I think) etc.
Some KDE versions are tied to certain KDE versions, the major KDE versions in use right now are 3.5 and 4.2. So it's best to install the appropriate package for your operating system rather than download it from the KStars website.
Ubuntu does, as stated above, default to the Gnome desktop environment, however you can install KDE through the Ubuntu package manager, you don't have to install kubuntu instead. Be aware that it will install 3-400 mb of packages if you do.
Finally, Stellarium (
www.stellarium.org) also runs well on the eee pc as does xephem.