What do you mean your site is no longer supported by CoffeeCup? CoffeCup is just an HTML editor, and it appears to still be available and supported on their web-site? Why not just download a version on your new computer, then use the built in FTP client to download your old site (assuming that it is still being hosted somewhere)?
There’s more to it than meets the eye. This is not your father’s CoffeeCup.
CoffeeCup has several current design tools, ranging from a sophisticated HTML editor to a modern Responsive Site Designer that uses frameworks from Bootstrap and Foundation Framer.
If this site was built with their old Visual Site Designer, it will be a bit of a chore to bring it back to life as-is, as there is no current tool that will easily open the original files. It would have to be done the old-fashioned way in an HTML editing program under manual control.
It can be done, but will take some time and effort...