It's a simple question but limited to how noun phrases function in English sentences.
Time phrases like last week, are an example of noun phrases functioning adverbially but they are still verb complements.
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Sure. Noun phrases can occur in an adnominal '-s genitive construction:
[the Queen of England]'s hat
Rarely, noun phrases can occur as the complement of certain unusual adjectives:
it was worth [the long wait in line]
conduct unbecoming [a good citizen]
None of the bracketed noun phrases ("the Queen of England('s)", "the long wait in line", and "a good citizen") is the subject of a sentence, the complement of a verb, or the object of a preposition.