The easiest way to shoot down the gunship is on Bandar Desert. Play as the Russian side. When the US spawns a gunship, shoot it down with the Pantsir-S1 AA Gun. It cannot be destroyed and easily shoots it down in a few bursts.
This is the easiest method because you can't be killed, you can't run out of missiles, and you do not have to know how to fly a plane to do it.
The developers have mover the AA gun back closer to the airfield since I wrote this. This makes using this solution slighlty more difficult. Just wait till the Gunship is in optimal range before you start to unload into it.