Given this liquid-liquid phase diagram, is the green line the nitrobenzene-rich phase?
Here is the same diagram with more annotation from the physical chemistry textbook by Atkins:



It shows that above the temperature $T_{uc}$, there is only one phase (the substances mix in any ratio). Below that temperature, there are two phases, with compositions shown by the solid lines. The horizontal line shows that if you try to make a mixture with a mole fraction of $a$ at the given temperature, you will get two solutions with mole fraction $a'$ and $a''$ instead. You can calculate the approximate ratio of the volumes using this technique. 


  Am I right in assuming that the right side (green line) is the nitrobenzene rich phase?


Yes, because the x-axis is the mole fraction of nitrobenzene, the phase with higher fraction of nitrobenzene will be that on the right, as you say. It does not necessarily mean that the fraction is higher higher than 50% on the right and lower than 50% on the left, though.