You got me thinkin'. How about this instead?
The prologue of the movie's the same. I think the prologue really works, so don't change a frame of it.
Rework the bit with Kirk as a boy to make it (a) less stupid, and (b) more clear that the premature death of his father has knocked him off his path to destiny. Maybe instead of getting into a bar fight with some ridiculously out-of-place cadets, he gets arrested or whatever and is given the choice between juvie and the Academy. He says juvie, 'cause he hates Starfleet for killing his dad, but Pike talks him into the Academy instead with an expanded "I dare you to do better" speech.
Flash forward four years. (We're about twenty minutes into the movie at this point, if that.) We're on the bridge of a starship. It's small, dark and cramped. As we dolly around the aft section, we see a silhouette against the viewscreen: It's Kirk.
"Captain, we're receiving a distress call," comes the voice from the communications station. Kirk's expecting this. "Kobyashi Maru, neutral zone," he says cockily. "Uh, no," says the communications officer on duty. "I mean, Captain, we're really receiving a distress call."
"Lights," says a gravely voice off-screen. The bridge lights come up, and the real captain walks into view. He's played by … hmm … Tom Waits, let's say. "Station, cadet," he barks as he passes the chair, and Kirk scampers. "Dammit," we hear Kirk say as he takes his assigned position. "I was gonna beat it that time." The officer next to him scoffs. "Sure you were, kid." Kirk spreads a grin you just wanna punch. "No, really," he says. He leans in. "See, I"—
"Helm, emergency speed to Vulcan!" shouts the captain from the other side of the tiny bridge. "Mr. Kirk, your repeat performance will have to wait." He thumbs one of those switch thingies in the armrest and gives the all-hands speech: planetary distress call, blah blah, ordered to respond, blah blah. We cut to an exterior shot as we see the small, old, unimpressive USS (I dunno) Parrot pivot, then streak off at warp speed.
Around Vulcan, it's chaos. The Parrot drops out of warp in the middle of a debris field, and in the distance we see explosions and flashes of light around the immense Narada. Cause the Parrot's so small it goes unnoticed at first, as the Narada targets the Essex and the Kiev and the Oberon and whatever the hell else. Captain Waits and his crew scramble frantically to get their bearings, and you can just see Kirk itching to get into it. He's not even scared, just wide-eyed with excitement. But Captain Waits wants none of it. "Helm, get us out of this mess and away from that ship!" he orders. The Parrot comes about and starts puttering away, trying to get over the horizon from the big fight. Kirk hates this, but he bites his lip. Meanwhile, the captain's giving orders to try to reach the Vulcan equivalent of FEMA or whatever so they can start rescuing civilians.
Just then, a missile from the Narada goes wide and slams into the Parrot's lower hull, tearing a huge hole in it and vaporizing dozens of compartments. The captain calls for reports from below decks, but communications are out. He orders Kirk to get down there and bring him back a sitrep, soonest.
Kirk bolts for the lift, but that's out too. So he ducks into the ladderway and slides down the (oh, let's say) four decks to the lower spaces of the little ship. He finds somebody, an engineer's mate or something (point being, an enlisted man, who are the only people aboard Kirk outranks), covered in soot and coughing. Maybe he does something heroic, like pulling him out of a burning compartment before dogging the hatch or something. The man tells Kirk that the hull's been breached, but the engines and life support are still operating. As he says this, the ship rocks again, and the lights flicker. "At least they were!" the man says. Kirk sprints back up the ladder to the bridge to report … only to find it smashed. The Parrot took a direct hit. The captain's dead, the bridge officers are dead or dying, the bridge is filled with smoke, the viewscreen is screaming red warnings at him. Kirk stands there for a beat, in shock … then he turns and runs for the ladder.
Below, he grabs the first person he can find. "Abandon ship," he says. "Pass the word!" "What?!" yells the enlisted man, disbelieving. "Get to the lifeboats, all hands!" Kirk repeats. Then after a second, "Captain's orders!" We see the enlisted man run for it, stopping along the way to pass the order, and Kirk heads back up the ladder to the bridge. There he clears the debris out of the way of the helm console and mashes buttons. The Parrot comes about, and the Narada looms large in the viewscreen. Little notifications start to pop up on the screen, one for each lifeboat away. When the sixth one pops up, Kirk sits back in his seat, a wry look on his face. "Yeah, well," he says to himself softly. "Like father, like son." He mashes buttons, and the Parrot lurches forward, on a collision course at flank speed.
We cut to a very wide exterior shot. The Narada is medium-sized in the frame, surrounded by clouds of fiery debris and burning disabled ships, but the Parrot is just a tiny speck. As the music swells, we watch the Parrot crawl forward, forward, forward … and disappear in a flash against the Narada's shields or hull or whatever.
And we hold on that shot for an eternity. Like three, maybe even four seconds.
Then the martial drums kick in, and we smash cut to a brightly lit and more to the point not-at-all-on-fire transporter room, where Kirk is just materializing. For a beat, everything is silent except for the fading hum of the transporter, and the drum-and-bagpipe thing on the score, and then a wall of noise explodes. We cut to Kirk's point of view. The room is filled with people in various degrees of disability, some limping, some laid out on stretchers. A female officer is screaming: "Clear the fucking pad!" (She probably doesn't say "fucking," but I love the image.) Kirk, utterly disoriented, gets out of the way just as another load of survivors starts to beam in. Another officer grabs him, demands to know his name, rank and section. "Kirk, midshipman first—I mean third lieutenant," he stammers. The officer ups-and-downs him, doesn't spot any arterial blood or protruding bones, and orders him to report to the bridge. (Does it really make sense to send him to the bridge? Of course not. But we need him there for the rest of the movie.)
And then … you know. Et cetera.