(By Celtic I mean the Indo-European people from Iron Age to Medieval Europe)
As far as I know the Celts had no written language (please correct me if I am wrong), but have there been Celtic songs or music that has survived? If so, what are some examples? And if not, what may be the closest thing or earliest example?
There are several depictions of musical instruments resembling pan flutes on bronze situlae from middle Europe like Vace and Certosa, showing their usage during festivities.
Actual musical scores that survived the ages... I don't know, but I highly doubt that. As you said, Celts had no to only very little written record and beside their rich archaeological heritage they left us to discover most of what we know about this Iron Age culture comes from Romans and Greeks and they liked to describe them as more barbaric as they (probably) were in reality.