How to pick the right capacitor for device with significant current peaks?
Over-design the capacitance required by pretending the power supply is too slow (but seriously, the pulse length of 1.8A is ~2.3ms, which your supply WILL be able to help with) and just look at an ideal capacitor and it's discharge curve, which is inverse log. 

To speed the calculation process, find some handy little online calculator like this:
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capdis.html

Given the requirement of 1.8A through the load, I am assuming you are running a 4V rail as given in the question, so I put in the parameters:


  Vo = 4V,
  
  R = 2.22 (so that I_load is 1.8A) 
  
  and after some estimation and a few values, I found 4700uF was perfect


A capacitance of 4.7mF means that without any help from the power supply, your rail will drop to ~3.3V after 2ms of the 1.8A load draw from a 4V charged cap. 

To fill in the gaps, you can safely assume the power supply will help out a little by then to reduce the rate at which the charge over the cap drops. You circuit may malfunction if the voltage drops below 2.7V or so.. 

Here is an example component that may suit the task at hand: 
http://www.digikey.com.au/product-detail/en/ECA-0JM472/P5118-ND/244977