I think your friend might be
Let me tell you 'bout a friend
from the age of ancient runes.
Possibly a reference to μ (which can be pronounced as 'me') or め or メ.
Also, a rune on the baptismal font in the church of Bridekirk (credit to the OP)
If you ever hear his name
you may think of songs and tunes.
"Do Re Mi Fa So" (sounds like me)
Though he lives in many places
— throughout Europe he is found —
I think the word 'me' pops up in several European languages including French and Spanish.
it's the land of Montenegro
where, enlarged, he's most renowned.
ME is the ISO 3166 code for Montenegro
When he visits Irish lands,
he's the same as what he owns,
I think this is a reference to the way an Irish person sometimes uses 'me' instead of 'my' e.g, "Where's me jumper?"
but the rest of his belongings
is among the ores and stones.
His belongings are mine, which is also a type of excavation in the earth. (thank you Rand al'Thor)
He begins to turn his head
The head of Me is M, which is W when turned.
When a cat gives out a shout.
Me + w = Mew.
NB The original intention of these two lines is that the beginning of "Meow" is "Me" which makes the subject of the puzzle turn his head. (credit to the OP)
You must tell me my friend's name,
and explain how you found out.