No, he doesn't. He's considerably old and has devoted his final years to helping the Vulcan refugees. He has very, very limited time to do anything.
No ship in their universe or time period can go up against the Narada, that was the whole point. Sending any ship back a few days to snatch Old Spock from the path of the Narada and slamming the Red Matter into it will only prevent the final few days we witness from happening, saving Vulcan, but leaving the entire 25 year altered universe in existance.
Sending any ship back to the Kelvin's time will add it to the casualty list, destroyed along with her, but doing little or nothing else.
Spock has to work with the technology of his time, even with his insight it would take years to develop technology soley from his memory, and that is already impaired and will only more rapidly become so, his age is a determining factor now.
And even if they did develop a weapon, return to January 4th 2233 and deploy it into the black hole, the Kelvin is watching a lightening storm in space become an anomaly from which a black squid from hell emerges, only to be destroyed by a futuristic Federation vessel. Far from repairing the timeline, this will only create a second, less violently altered one, but an altered one anyway.
So really, why?