It's definitely not standard and I don't recommend using it generally, including in songs. Replacing "I" with "me" is something which we have heard people do in Caribbean countries but I have no direct experience of this - it's just a common belief. It's not clear to me why the Beatles did this in this song: perhaps they were imitating a specific person they knew who used this phrase.

I don't think this has anything to do with Jamaica or regional dialect. It helps to read the lyrics in their entirety. This song is a confession by a young man who is seriously immature and has a history of beating his girlfriends (John Lennon said that he was responsible for the reference to beating women, and that it was based - remarkably - on his personal behaviour). The young man's belief that things are "getting better" - which is the title of the song - is doubtful at best. The lines: "Me used to be angry young man/ Me hiding my head in the sand" do a great job of emphasising how self-centred (me, me, me) and childish ("Me happy" and "Me angry" are baby talk) this guy is. They also work with the song's meter (rhythm) in a way that grammatically correct constructions don't. The original question was "I am wondering how common such a use of 'me' is." The answer is that it is not common and that it is illiterate, like the young man in the song. McCartney and Lennon would never have spoken like this in their own speech.

I wonder if it's the same influence that caused McCartney to write "life goes on, brah" in another song a year later? Yes, it's typical of Carribbean English (which often spills over and back into the creole languages) - "mi" is used in place of "I" in Jamaican Patois. The British children's series "Rastamouse" uses this style of speech often.

I think "Me used to be angry young man" is a short form of "It's me who used to be angry young man". And I think you can use this in daily conversation or songs :]]