I've pasted some images below- the first is just the masks of what was being measured, the second was the DAPI measured in the nucleus, the third is the DAPI measured in the cytoplasm. I made this in the WorkspaceViewer. You could do something similar too for your Ratio.
One thing that I noticed in your pipeline that's pretty problematic is the ReassignObjectNumbers module- I think you're doing that to try to merge the DAPI nuclei in each cell, but it's set wrong, as evidenced by the fact that some cells below have two nuclei measurements:
A) the parent object should be the cell, not the edited_dapi_nuclei (which is acting as both parent and child)
B) you don't want to use the Convex hull around them (which would include anything in between them if they're not physically touching) but instead would want to use the Disconnected method. In any case, unless you have a large number of binucleate cells you can't bear to lose, I can't help but think that using the ReassignObjectNumber module will promote some weird behavior- not least of which because ReassignObjectNumbers doesn't ensure that the Cell and its corresponding Nucleus will have matching object numbers. I think you'd be far, far safer instead using FilterObjects to keep cells that have 1 and only 1 nucleus, and using its 'Relabel additional objects to match the filtered object?' setting to make sure object numbers are matched between Nuclei and Cells- that's the best way to ensure your ratios are calculated correctly.
As for whether or not you should be using integrated intensity, I'm not really sure; it depends on what you want to know. Mean or median will give you the concentration in each compartment, Integrated will give you the total in each (which will be heavily dependent on area); which one is "correct" depends on your question. Yeast also tend to be pretty autofluorescent, so you may want to try to do some sort of background subtraction (either in the image, or to "zero out" the mean or median values after the same way you'd do on a standard curve on a spectrophotometer) if you want to see the sort of difference between stained and unstained regions you'd expect.