Translation:I do not listen to a lot of music.
Amari + negative verb equals I don't really .....(whatever the verb is). Amari is an adverb so you wouldn't translate it as I don't listen to a lot of music because that is referring to the quantity of music you listen to and for that to be correct amari would have to be an adjective (cos it would be describing how much music you listen to). But amari is describing the verb, not the noun and so is referring to the frequency of the speaker's music listening.
So "ha" (which in these cases is more similar to a "wa" sound) identifies grammarly the main subject or topic of your phrase or comment. "wo" would highlight you're talking about a thing or object more in the way of a direct object sense. You could still use it, but it would be different.