It's a little scary that this is so old, but it solved a problem I was chasing myself.
Yes, the Arduino can run this display the same as the backlit 2x16 LCD display, since the display actually has a controller built in the back of the display. Thankfully, this unit even came with a wire we could use.
I think I bought mine at the same time, same place, but until I started putting a microcontroller class together, never really sat down and worked with it...
using the same 4-bit method should work, and this display was the same as the Kronos 4500 time clock, and is at least 2-color with a green backlight.
I expect to get it working for my project shortly, but if you are still looking, I'll post the information on this here soon.