Alligator popup is a simple plugin designed to open links in a new browser window. A lightbox is not the same as a new browser window, rather it is an overlaid div within the same browser window - a totally different concept with a totally different implementation.
If you need a lightbox, alligator popup is not an appropriate plugin to use. It would need to be rewritten from scratch to support this and it would be a totally different plugin which no longer does what Alligator Popup was originally intended to do.
To be able to tell you how to use the code you linked to to create a WordPress plugin with a mode of similar usage to my Alligator Popup plugin I would need to first check the code and then create the plugin and test it fully to be sure that it does work as it should. I'm afraid this is far beyond of the scope of support for the alligator popup that I can provide.
I would only be able to make such a plugin if I needed it for a project I was working on or if someone commissioned me to created it.