Say I have three unicellular organisms: a eukariote, a bacterium and an archaeon. If I cut off nutrition from them at the same time, how long will it take for them to die? What will their death look like, and do they actually need to die? If so, why?
This is very dependent on the organism within each of the groups you mention. While for the most part, archea are the extremophiles and have the ability to withstand many extreme conditions, nutrient limitation survival greatly varies. I think you could easily find organisms in each group that could withstand nutrient limitation well.
A good example would be yeast, which is a single celled organism you may have actually worked with. If you take laboratory yeast, in their diploid state, and starve them, they will undergo meiosis, and in their spore form, live for a very very long time without nutrients, protected by their ascus (which is actual their dead mother cell).
Yeast can also completely desiccate and still live. When you pick up an active yeast packet for baking, those yeast have not only been starved, but have been completely sucked dry of their water. Yet when you add water back, they live (though a great number of them will die).
I think in general, most single celled organisms have a certain tolerance for low nutrient conditions. This does happen to them a lot. All it means is they slow their cell cycle down, stop dividing, and just undergo a caution period. There is all sorts of signaling in the cells that sense the nutrients, and slow growth based on their conditions.