I am aware that Absolute pressure is zero-referenced to a perfect vacuum and Gauge pressure is zero-referenced to ambient air pressure. So, Absolute pressure is equal to Gauge pressure plus atmospheric pressure.
However, the Part B of this problem, confused me on what the absolute pressure under piston B is, which comes from the fact that the system is open on both piston A and B. In this case, is the (Weight of block)/(Area of piston), Gage or Absolute pressure?
Your explanation of the differences between Gauge and Absolute is correct. As you have noted, the difference between them is due to the pressure exerted by the atmosphere.
The pressure exerted by the blocks, therefore, is a "Gauge" pressure, and must be added to the atmospheric pressure (dependent on altitude, etc) to find the Absolute Pressure in the fluid. It is important to note that The pressure exerted by the blocks is independent of atmospheric pressure - it would exist in a vacuum.
It doesn't seem that the test creator relies on your ability to differentiate between absolute and relative (or gauge) pressure. It's not mentioned anywhere, other than in B where he wants you to give the absolute pressure, which is impossible to answer, since no ambient pressure is given.
It's not really possible to give an answer at all since the described situation is not static. If the mass is increased to $150\ kg$, it will launch piston 1 rapidly in the air. Furthermore, in a static situation, the pressure below B is/must be the same as below piston A or anywhere else in the fluid. The whole question gives me the idea that the creator of the test doesn't really know what he's talking about.
I think he just wants to hear what the relative pressure in the fluid would be, if the mass on piston B is $150\ kg$, and piston A is frozen. Then you would come to $3316\ kgf/m^2$ or $32.1\ kPa$ relative, assuming a gravitational acceleration of $9.81\ m/s^2$.
The absolute pressure doesn't matter anyway, since only relative pressure will be able to do any work. You can mention that the ambient pressure should be given to be able to give an absolute pressure, but in my experience, smartassery is rarely rewarded in school. Although the teacher is just plain wrong here and he should accept his fault...
