It is easier to see how these phrases are functioning if the sentence is structured in a more standard way, without ellipsis (shown in parens).
Music, once admitted to the soul, becomes a sort of spirit, and never dies.
Music becomes a sort of spirit, and (it-spirit) never dies, once (it-music- is) admitted to the soul.
The main clause is: Music becomes a sort of spirit.
To more directly answer the original post, the 'phrase' in question is neither a phrase, nor a conjunction. It is a subordinate elliptic adverbial clause. This clause is functioning adverbially to establish time relations to the other verbs. Viewed in the standard sentence order (above) it modifies the second clause's verb 'to die' as a time sequence, and the first clause's verb 'become' to say when the action happens.
The word 'once' and the clause itself are not modifying the subject of the main clause (as suggested in another answer), at least not in my opinion. Adverbials modify the meaning of verbs, not nouns. Music is the subject of the first and last clauses, and spirit is the subject of the second clause. At the clause and sentence level (S+V+C) there are grammatical relations between the subject, verb and complements phrases that do produce meaning but it is not an adverbial modification.
The words and and once are subordinating conjunctions as DCShannon has already indicated. Mostly because of the ellipsis, the other two clauses are fragments or dependent clauses.