Classic environment means pre-OSX. Generally OS 9. So no, the OSX LW will not run in OS 9. You need one of the more recent LW versions depending on your version of OSX. You are obviously not running an intel mac, so you would be unable to use LW version 5.
See which version suits you here: Lime Wire » Download Out-Of-Date Installers
For OS 9, choose the classic version.