Downstopper

When do we use a downstopper?
The term "Downstopper" is not a term that is widely used. In fact it is not used in my country at all. Because your next question makes reference to a vertical vessel I am going to assume that you mean "Pipe Support"

Where should we place a downstopper on a vertical piping?
The pipe support for a vertical pipe line on the side of a vessel shall be located as close to the highest point on the pipe line as possible or the top tangent line of the vessel whichever comes first (in the up direction). The support may be a twin cantilever attached to the vessel and a Trunnion attached to the pipe. The twin cantilever support attached to the vessel shall be provided by the vessel fabricator and shall be designed based on the live loads and dead loads furnished by the piping department.