Q:

It is May 12th, 1959, when I'm introduced to Janey

Is it possible to use the present tense along with an adverbial of the past time?

(1) It is May 12th, 1959, when I'm introduced to Janey.

This is a transcript of the movie "Watchmen." And you can hear the movie sound here.
Is it really natural or even grammatical to use the present tense "am" along with the adverbial "May 12th, 1959" as in (1)?
Since it's a cleft sentence, you should be able to change it to:

(2) I'm introduced to Janey on May 12th, 1959.

OR

(3) On May 12th, 1959, I'm introduced to Janey.

If (1)'s natural, how about (2) and (3)?
EDIT
I'd like to make it clear that I do understand the use of the present tense for the past time in a narrative setting. The question really is not about the use of the present tense itself but the use of the present tense along with the past-time adverbial. 
Is it okay to use the present tense along with the past-time adverbial when you describe the past time in a narrative setting?

A:

You need to understand that "Watchmen" [sic] is a film with a very particular mood throughout. Although set in the past, the language throughout is immediate; present tense. The author is trying to bring the audience into the time; not through flashbacks or memories, but as though you were there, then.
The whole mood is retro, but even more so. It's very powerful in that respect (not to mention many others...)

A:

Yes, it is okay to use the present tense along with past-time adverbials when you describe past time in a narrative setting. It is the historical present. It can also be called the storytelling present and narrative present. Jokes also often  use the present tense. The use of the present tense gives the sense of immediacy. 
Such uses can include adverbials of time (which talk about past time) treated in the present tense:

Yesterday, John goes into a bar. He orders a drink. He begins a conversation before leaving. He forgets to pay.

Source" As Time Goes by: Tense and Universal Grammar - Page 199

"It is a bright summer day in 1947. My father, a fat, funny man with beautiful eyes and a subversive wit, is trying to decide which of his eight children he will take with him to the county fair. My mother, of course, will not go. She is knocked out from getting most of us ready: I hold my neck stiff against the pressure of her knuckles as she hastily completes the braiding and the beribboning of my hair. . . ."

Alice Walker, "Beauty: When the Other Dancer Is the Self." In Search of Our Mothers' Gardens: Womanist Prose. Harcourt Brace, 1983. (Cited here.)
and

Two weeks before Christmas Malachy and I come home from school in a heavy rain and when we push in the door we find the kitchen empty...

from Angela's Ashes, by Frank McCourt (cited here.)

