According to the rules of commas, and there are many, additional information that is not necessary to the meaning you wish to convey is set off by a comma. I'm simplifying the language of my explanation for any reader who is not a native speaker of English.
Would the coming Tuesday at 12:15 be okay for you?
In your sentence, there is no comma required because all of the information in the sentence is necessary to convey, to pass on, the meaning you want. All of the information defines the meaning: the time; the date. None of the information is additional information, such as information that defines the noun more (an appositive or a non-defining relative clause).