I would use 'my". You are talking about "your first time".
I would write the sentence as: "My first time hearing this song was in 2003; it was on Alice's CD player.
Student or Learner
firstly, I have a problem on distinguishing the "the" and "my"
Secondly. I don't know how to add more content after the phrase of my first time.
I was trying to say: My(or the?) first time of(is the of needed?)hearing(hearing or heard) the song was on (or from?)Alice's CD player in 2003.
is this sentence correct?
Or more simply: I heard this song for the time in 2003 on Alice's CD player.
Or: The first time I heard this song was in 2003, on Alice's CD player.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.