This is just an alternative and less aggressive way to getting this achievement. It is not faster, but does not require any work, and is for music pass holders.
Just search for a band with a lot of songs that you like. I listen to like Florida Georgia Line, Luke Bryan, Thousand Foot Krutch, Shinedown, Disturbed, Linkin Park.
I started off by listening to many bands during the day while playing games (snapped the music app), and at night, I would fall asleep to Linkin Park, and their playlist was big enough to last until I woke.
Later on, I created a playlist of all the bands I named above, of almost every song they had, so it was just a huge playlist of a large number of songs.
I would just let the playlist play through to the end. My current playlist ends after 9 days of non stop running it from start to end. My internet friends tell me that they max out at 100 songs, but I do not know why that is. I pay for a 1 year Xbox Music Pass ($100 for 1 year of unlimited music). I make my playlists on my computer, and I have not run into a max size yet.
But yea, doing this, you can literally just leave the app snapped, and it will play music. I get any where from 300 - 1000 songs a day doing this (I might be playing a game during the day and the music might not match the mood, so I skip some songs sometimes). This method speeds up the process of getting the achievement, but does not require additional effort after the first time of making a playlist, as you are just playing music through games and to fall asleep, etc...
Also: If you run into a cut scene in a game or something, just unsnap the app using Smart glass if you hate Kinect, or if you are like me, just say, "Xbox Unsnap" and it will pause it where it was. When you are ready to play music again, just snap it again.
The controller can be turned off. It will not affect anything. You can also control music through the Xbox One Smart glass.