Me la paso pensándote
This is more like a figure of speech. A rather pseudo figure of speech IMO. Now, me refers to the person aforementioned. In the song he is already talking about himself before so he can refer to himself again by only using Me.
La is the singular feminine of a thing. This means that it refers to something that I am talking about, in this case waiting or "being" as paso means just thinking in some way. This is why this is a sort of figure of speech.
In a nutshell, Me la paso pensándote means that he is thinking all the time about this person. The person he is talking to. Since paso is being used as a verb that denotes constant existence or continuity of an action.
I know this can be very confusing I hope this has made it clearer.
Here is an extra resource about how this is constructed although I personally think that this is a very special case of this kind of statement.
Me, te, se, los seguidos de lo, la, los, las