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Isn't "up" an adverb in that sentence?
No. Up by itself is a preposition. The adverbial form is upward. Also, it does not answer a question about the verb. "Where are they shooting? They are shooting 'up'." This makes no sense. That would mean that they were firing into the air, like celebratory gunfire.
"To shoot up" is a compound verb, with the prepositional addition differentiation it from the verb "to shoot." You won't find this in a lot of English grammar books influenced by Victorian-era Latinization, but it is well documented in English's main parent language, German.
In German compound verbs, the preposition is included in the verb in the infinitive form and separated and put at the end of the clause when the verb is conjugated. For example, "ziehen" is "to pull," and "um" means "around," but "umziehen" means to move (like to a new house), not "to pull around." When you conjugate it, "um" comes off and gets put at the end. Therefore if I want to say "Bob is moving to Peoria," it would be "Bob zieht nach Peoria um."
In the case of compound verbs in particular, ending a sentence with a preposition is actually more natural for English, a Germanic language, than trying not to in order to follow Latin's example.