I don't think PNG will let you have georeferencing within the file itself. A format like GeoTIFF would include georeferencing. PNG won't. The best you can do is have what I would call a 'sidecar' file - such as a .wld file to specify georeferencing for that file.
If you move the image to the correct place in ArcGIS, I would assume it's creating some sort of referencing of this type.
If you want some software to do this for you, then I use FME (though I do work for Safe Software who develop it). It will read Shape data, let you rasterize it, then write to PNG (with a .WLD file to act as georeferencing).
Of course one other question is what you expect to see in the PNG. Since Shape data doesn't contain any idea of feature colour, what colour will you expect to see in the PNG?