Is the phrase, “Can you open me up a bottle of wine?” grammatically correct?

(Frank Dauenhauer, former Technical Writer & Editor of Company Publications at Eastman Kodak Products and Services (1960-1991)

Yes, it is correct. The sentence is using the objective pronoun “me” in the dative case,¹ which means “to me” or “for me,” but the preposition “to” or “for” is not enunciated but is understood. So the sentence actually means “Can you open up [for me][to me] a bottle of wine?”
What is the Dative Case?: A noun or pronoun is in the Dative Case when it is used as an indirect object.

Ex. Oma gave me a puppy.
This sentence contains two objects, a direct object and an indirect object.

To find the direct object, find the verb and ask “what?”
Question: gave what?
Answer: gave puppy.
Puppy is the direct object. It receives the action of the verb.

To find the indirect object, find the verb and ask “to whom?” or “to what?” “for whom?” or “for what?”
Question gave to whom?
Answer: to me
“Me” is the indirect object.
“Me” is a pronoun in the dative case. It does not receive the action of the verb directly, but it does receive it indirectly.

Here are some more examples of sentences that contain nouns or pronouns in the dative case: