The portrait fell down is that an advreb or preposition phrase

It is an adverb phrase.

Explanation:

Adverb:

It modifies a verb, adjective, another adverb or group of words. In this example adverb down is modifying verb fell.

Preposition:

Typical definition of preposition in most of the dictionaries/online resources include that a preposition is used before non, noun phrase, pronoun or -ing form of verb, And they usually show a relationship between two nearby words.

In this example, there are no nearby two words between which down has to show relationship. Let's complete this sentence, for instance, the portrait fell down on the table, or the portrait fell down in the tub. We have to use a preposition on or in to show portrait's relation with table or tub. This shows, since there is no word to show portrait's relation, so no preposition is used.

yes, it can be both.

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