The mode is melancholic Mixolydian with 2 sharps and the tonal center around A.
I have made a video and an arrangement of this song for organ which you can watch and print out for playing here.
The thing is that nobody knows for sure how it supposed to be performed (about 2000 years have passed since its creation. So this is one possible version. Feel free to change anything you want to suit your needs (texture, registration, key , add pedals etc.)
The sounds of this video remind me of Greek hydraulis. The score is very easy - even complete beginners will be able to play it. I intentionally wanted to keep the texture straightforward because presumably that's how it might have been played on hydraulis (and sung of course - here are the words and the full description of the song).
Enjoy and share!