If I have a .022" diameter cylinder that is 3' long laying horizontally with water flowing through it at a rate of 80mm/min, What is the pressure in the tube?

Draw a straight line between the inlet pressure and the exit pressure will give you roughly the pressure distribution along the tube from the inlet to the exit. Assuming that the flow in the tube is fully-developed flow, the pressure gradient along the tube should be a constant ( you need that pressure gradient to overcome the frictional force on the tube wall. The longer the length of the tube, the larger the pressure difference between the inlet and the exit.