Mailing complete. The rest is up to you, postman.

Literal

Posting completed, [quotative-と]. Remainder [topic-は] entrusted [emphatic-ぞ], postman [vocative-よ].

投函完了 ('posting complete') is a self-narration of an action just performed — と at the end marks it as inner monologue. 後は頼んだぞ ('the rest is up to you') uses 頼む ('to entrust/request') in past tense for immediacy. ポストマンよ uses よ as a vocative particle, directly addressing the postman. Playful, self-aware narration — the speaker dramatizes a mundane act.