There are a lot of things to unpack here. If you only want the notation to a song, ask someone to do it for you or purchase the sheet music. If you want to develop the skill to transcribe yourself, reverse the process and train your ear:
I suggest you visit your local Protestant church and steal a hymnbook. Or, ask to borrow one. Then each day, sight sing the S, A, T and B parts AWAY from the piano to one or four hymns. Did I mention you HAVE TO DO THIS AWAY FROM THE PIANO? Hymns are pretty easy to sight sing. It may be hard at first but it will get easier. By the time you get through half the book you will start to notice that when you hear music, you will just know what the notes are.
For example, in the standard hymnbook, look for ODE TO JOY and you would sing for the Soprano, 3345543211233 22. For the Alto, probably 5555567855578877 and for the Bass, 888855543321555.
You may want to go to a Protestant church that still sings hymns and during the hymn, sing the S for a verse, A for the next, T for the third and B for the fourth. If the choir or soloist sings something, find the One from the introduction and try to figure out the melody. Again, it will be difficult at first but will get easier.
As far as hearing what a blizzard of notes are, study music theory. By knowing what scales, chords and arpeggios sound like from the aforementioned ear training, you will hear fast notes and just know what they are. But you can't be satisfied with just Ionian and Aeolian modes. You have to study all the modes and other scales such as whole tone, chromatic, pentatonic, diminished . . .
Above all, sing EVERYTHING. Play one of the aforementioned scales or modes and SING them to get them in you. Get them in your voice, brain and ear. Then when you hear them, you will just know what they are. Just like hearing someone speak a foreign language and knowing what it is because you trained your ear to the dialect. ¿Comprender?
Another analogy would be like studying Latin. Then when you hear a word you never heard before, you will just know what it is because you know Latin roots. For instance, aqueduct = water + to lead. Air duct = air + to lead. Abduct = away + to lead. Adduct = toward + to lead. Like spreading your fingers out and back.
Knowledge is power and music theory is the vocabulary and alphabet of music. Not just about dots on a page to be matched to a key. That is just dog and pony tricks. Remember, shortcuts only cheat yourself.