Once (after or when) I get home, I get hungry: In any case, all three of these words are acting to join the two clauses, but it is a bit easier to see the functions of these words when the sentence is in standard order (S+V+C):
I get hungry after (or once or when) I get home.
It should be pretty clear that these all function as adverbial conjunctions (aka 'conjunctive adverb' or 'subordinating conjunction') to join the two clauses with a time relation. Words are just words, most of the time they can't be typed until they are used in a phrase or sentence. Accordingly, prepositions are always prepositions, but some words like 'after', 'when' and 'once' can have different functions (types), depending on how they are used.
The hard rule about prepositions is that they are always followed by nouns. But 'once' never functions as a preposition, and 'after' in this case, isn't one either, even though it is followed by a noun. Prepositions never bridge clauses, they are always internal. Another rule is that they establish a meaningful relation between the noun that immediately precedes them (or a preceding verb phrase in their clause), and their object noun. In this sentence all three words create an adverbial relation between the clauses, none of them are functioning as a preposition. They establish a time relation (sequence) between the clauses, that's all. They are not adverbs because the have only an indirect relation to the verbs in either clause.
In this sentence, 'once' an adverbial conjunction.
Update 01/2019: Maybe once can function as a preposition.
How about this sentence.
I talk to my sister once a week. S+V+AC(pp)+AC(pp)
Once seems to be working as a preposition in this case. It is creating a relation between the verb and its object a week. It definitely has an object and that relation determines the meaning of the sentence. This use requires the present simple - We can't say, * I talk to my sister once, it would have to be in another tense to be an adverb and have the meaning one time.