Hmm. In the interest of helping each other out, since this is a cooperative game, I'll share my intel.
The US has a carrier assigned to the defense of New York, unsurprisingly. Though I'm not certain of the details, this means they will be sending a decent number of jets to attack the Kaiju. Presumably, they will be modern jets--better than USSCCI's own last generation jet--and will be able to do quite a bit of damage. Whether or not they will be capable of killing the kaiju, I don't know. They'll certainly be able to do quite a lot of damage to it, and if it doesn't die, it'll be much easier for our forces to kill it after the US jets start their runs.
The runs will begin four or five turns after landfall, and will likely be sustained for a few turns after. Functionally, since there's a monetary award for damaging the Kaiju, this means that there may be a kind of damage race; hurt the kaiju before the US fighters kill it. On the other hand, hurting it will be easier after the US fighters start dealing damage, so if your units can't be guaranteed to survive five rounds, and you aren't sure they'll be able to chip it, having some stay back until the kaiju is weakened might help ensure that you collect the bounty. I'm genuinely unsure.
As for not-hitting-buildings tactics, I simply don't know what's best there.
If anyone is willing to share their secret information, it would be much appreciated.
I'm deploying the assault fish, artillery piece, infantry platoon, and combat jet. Equip the combat jet with dual warhead missiles.
The artillery piece and infantry will be under orders to engage with the kaiju immediately. They are to not risk hitting civilian buildings, until the first US fighter wave hits, after which point they should take every shot available. EDIT: The artillery piece and infantry will take every shot available to them, regardless of risk of hitting civilian targets. The assault fish is under orders to hold fire until it's basically within close range, and then to fire everything and not let up. Finally, the jet is not to even enter range until the Kaiju has been engaged by other units. Only once it has been engaged will the jet enter the battle. The jet, once entering the battle, should stay at max range until it has a clear shot, or if it cannot get a clear shot, until turn 4. After it has begun to fire, it will take every shot available to it until it has either been damaged or expended all its ammunition, at which point it will attempt to retreat.