Which Capacitor I need?
If we go to the Cap equations, we have the relation between the capacity and the current

If we resolve C, then we have:
This means that, the capacitance will be = C = I*t/(ΔV). If you have 2A peak, and following the SIM800L the time of the burst is about 600uS, and the maximum voltage drop from 4V is 1V, looks like the min voltage for SIM800 is 3V, then you have you will need 1200uF, but this is not taking in account the ESR of the capacitor..., so this is conservative calculation, and only taking in account that the capacitor is the only which is giving this current.
That is why, super-caps are used in this applications, or stronger power supplies.