Knowing that solubility of NaCl is 35g/100 mL and sucrose 182 g/100 ml, which one would precipitate?
I found data on sucrose and glucose, shown below.
Source: https://pubs.rsc.org/en/content/articlehtml/2017/fo/c6fo01497f
At room temperature, the binary water sucrose mixture can have up to 58 volume percent sucrose, and the binary water glucose mixtures can have up to 44 volume percent glucose. If you try to dissolve both at these levels, there would be no water left.
For some supersaturated solutions of glucose and sucrose in water (e.g. 60% sucrose and 30% glucose), you can have either glucose (dashed line) or sucrose (solid line) precipitate. Which one comes out of solution would depend on the kinetics of the two crysallization processes.
What is the effect of adding another solute to the solubility of the initial solute?
For ideal solutions, there is no effect. You could explore this with two solutes of extremely low solubility (because with low concentrations, solutions approach ideal solutions). So adding silver bromide to barium sulfate should not affect its solubility. This example works because the salts have very low solubility in water whereas silver sulfate and barium bromide have moderate solubility.
You could also use two molecular solids as example, e.g. caffeine and aspirin in cold water.
The reason there is an effect in the case of sucrose is its high solubility. At the high concentration, neither sucrose nor water behaves ideal (e.g. their activity coefficients are far from 1).