Lost small screw in sink, can I open tubing to retrieve?
That style of sink drain that you show has a trap at the bottom. A trap on a sink, regardless of the style you show or the traditional U style, serves two purposes:


Catch things that people drop down the drain
Prevent sewer gasses from coming up


This means that any water device that has a drain in a house must have some trap. IF your sink did not have the one shown in the picture, chances would have been that it is down stream somewhere.  In both styles, you simply can unscrew where it attaches with a wrench of appropriate size and the trap comes off. 

Note: Keep in mind that after retrieving your item to let water flow through it for a little bit again as this puts a layer of water back in the trap which is what really protects from sewer gasses.