StarTropics III only runs on Windows. If you don't have Windows then you are out of luck since there are no intentions of making it cross platform. Everything has been tested to work on 32-bit and 64-bit versions of Windows XP, Windows Vista, and Windows 7.

The download comes bundled with the newest versions of both the game and the level editor.
StarTropics III Setup

In order for the game to run you must do a one time installation of the Microsoft .NET Framework 2.0 or greater. If you already have this installed then there is no need to download this. If you are unsure if you have this installed or not, you can check to see if it is already installed on your computer by going to the Control Panel and opening the Add or Remove Programs (XP) or Programs and Features (Vista, Win7) in order to look for it.
Microsoft .NET Framework 4.0 from StarTropics HQ
Microsoft .NET Framework 4.0 from Microsoft.com

The download contains updated sound effects for ST3 created by Cleveland Rock. To use, simply unzip and copy the contents into the sfx folder in your StarTropics III directory. NOTE: These will replace the original ones that come bundled with the demo. You'll either need to make a backup of this sfx folder or redownload the installer again to get the original sfx back.
Updated StarTropics III SFX by Cleveland Rock

The file format for how dungeons are saved changed with the release in September 2010. Anyone who has any existing dungeons that they want to work in the new editor will have to download the converter. You can tell if you have any old dungeons because your StarTropics III directory will have a customDungeons folder. The new location for where dungeons are stored is simply the dungeons folder.
Custom Dungeon Converter