Q:

thought experiment involving water pressure

If I were to move the syringe up, and cause it to be stationary, how does that effect the pressure at the bottom of the pipe? How does the movement of this water effect the system setup and pressure? 

A:

Since it is closed at the bottom, the pressure felt at the bottom of the pipe is a function solely of the weight of the column above it. So this is the combined effect of the depth of the water column and the depth of the air column above.
So, if you move the syringe up, it will displace the water from the spigot area into the main pipe, making the water column deeper. So pressure at the bottom will increase.
Conversely, if you move the syringe down, more room is available in the spigot and the water column gets smaller. So pressure decreases.

