I feel like this sentence is grammaticaly incorrect when saying it but I can't find the mistake, or I feel that i could put "at" at the end. So, what is the right thing? Is it the original sentence, or is there is really a mistake, or can i put an "at" at the end?
The sentence is grammatically correct.
I still remember the first time I listened to this song at.
Would not be correct. You could feasibly say:
I still remember the first time at which I listened to this song.
But it is not really idiomatic in this context. You sentence is fine as it stands. You don't need to use at when you are talking about an occasion rather than a clock based time.
The sentence is grammatical, but I think you're getting confused between two different meanings of time.
The basic meaning is "time on a clock", and at is the appropriate preposition for that meaning.
But in sentences like this, it has a different meaning, "occasion", and we would not use at with it. If it stands as a prepositional phrase, it uses for ("I have just listened to it for the first time"); but in this construction, it doesn't need a preposition at all.
[If you meant "time of day", you would have to make that explicit: "I still remember what time it was when I first listened to this song"].