No, ne without pas doesn’t mean “only” – ne … que does.

So the translation is “I’m rich only from my friends” (I’m rich only thanks to my friends, thanks to friendship.)

I would translate the sentence as:

My only wealth is my friends.

It is not possible to add pas in the sentence Je ne suis riche que de mes amis.

Note that the ‘ne’ in this phrase is not technically necessary, and since you say it comes from a song, might only be there for artistic reasons.

Je suis riche que de mes amis will be understood the same.

Additional note: both forms are a little bit stilted and contain some measure of poetic license. Drawing ‘rules’ from such phrases is not the best idea.