Sink clogged completely when using NaOH drain cleaner
How to unclog it

!!! WARNING !!!
Put on gloves, and full face protection gear! NaOH can cause severe burns or even permanently blind you. You wouldn't want even a droplet on your eyes, since (unlike your skin) they have no protection at all. 

(Before trying my method below, give a try to the other suggestion by @Jaroslav Voller. Try not to pierce the pipe if you use a screwdriver, though.)

Take 2-3 straws and insert each one into the holes. No I'm not joking.



Then pour hot water through them using your tap. The hotter the water the more soluble NaOH and the salts you created, will be. Which will make the whole process faster. Just let the water run until it is unclogged. 

Be careful with this water since it contains large amounts of NaOH and is corrosive. 

This will remove the NaOH layer, but your initial problem will probably not be solved so you will have to try using the drain cleaner again, but this time use less NaOH.

Why it happened

The grid at the entrance of the pipe reduces water flow. My assumption is that the excessive amount of NaOH you used, along with the reduced water flow right after, caused the NaOH to boil instantly the small amount of water that was already present and formed the solid NaOH layer which you described. 

Pouring water doesn't work because the water already present, along with the grid, reduces the flow above the NaOH layer, meaning it will dissolve very slowly or perhaps not at all. 



When you started throwing the citric acid, it dissolved in the water and reacted with the NaOH, increasing the temperature locally and also creating a sort of "salt", sodium citrate. 



Initially it was soluble but as soon as it's concentration increased and the temperature dropped, it's solubility dropped as well, forming a layer of salt (blue color in the figure below), which actually separated your acid and base. This is why they stopped reacting. 



Using the straws to guide the water flow directly on those layers will do the trick. The layers will be dissolved within a few minutes. 



Note: Colors in the figures do not represent the actual colors of the substances used.