I am a prime subscriber and cannot get prime music through my echo
The short answer is you can't have access to the Music Unlimited,... I was also under the impression that we could play music via Amazon if I was a Prime Member, however, Amazon wants even more money from you if you would like to play their music, I was really, really disappointed when I discovered this, even still, I'm in to minds of canceling my prime.
Moving on, If you download the Amazon music installer and uploaded your library you can play your own music via Alexa, furthermore, go to your Account on top right and click the drop-down menu and look for "Your Music" (Click, Your Music) Once there, do a search in the search bar at the top and you'll see some albums with a + sign,(When you Mouse over them) these you can add to your library to play via the Alexa device.
Good Luck