Some water will evaporate, and the heat required for this comes from the water itself. The water will cool down a little (latent heat of evaporation is a function of temperature, going from about 45 kJ/mol at 0 °C to 40.7 kJ/mol at 100 °C). The heat capacity of water is about 75 J/mol K. This means that evaporating one gram of water would extract enough heat from 600 gram of water to cool it down by one degree Celsius.
Evaporation is a really effective way to extract heat - it's why sweating works...
At a given temperature, in your liquid water-air system, equal numbers of water molecules will enter the air from the liquid as return to the liquid from the air. The system will be in equilibrium and the air will be "saturated" with water vapor.
There are two ways that condensation will form on your ceiling. If the air is supersaturated with water then your "nucleation" sites will facilitate condensation. But conditions in this system are not those that would result in supersaturation. The second way for condensation to occur is for the ceiling to be colder than the air.
If condensation occurs on the ceiling, much of the latent heat of condensation will be transferred to the ceiling therefore warming it. To continue the condensation process, you will have to keep the ceiling cool requiring expenditures of energy from outside the system. Your system is closed but it is not isolated in thermodynamic speak.
Furthermore, as the condensing water loses heat to the ceiling, the system cools. This will result in a lowered equilibrium vapor pressure, that is, less water in the vapor state. To make matters worse, the lower temperature of your system will require an even greater lowering of the ceiling's temperature to maintain condensation.
As far a entropy, you must, in addition to events within your system, consider those happening outside to power the refrigeration process.
Hopefully you understand the turbine you may be running inside the system won't even come close to powering the refrigerator outside!
Best Answer
No, in fact you could even view the spontaneous evaporation as being driven by the fact that it increases entropy.
Basically what's happening is the liquid particles have random speeds (with distribution characterized by temperature), and they bump into each other. Every once in a while, two particles near the interface will collide in just such a way that one of the particles will escape the liquid and transition to the gas phase. The particle that remains in the liquid will loose energy in the process, thereby slightly lowering the temperature of the liquid. However, generally a gas has considerably more specific entropy (entropy per particle) than liquid at fixed temperature/pressure [1]. So, the overall entropy of the system increases.
[1] Compare for example the molar entropy for solid, liquid, and gas phases of water in these tables from Wikipedia.