Does vapour pressure of a liquid solution depend on the size of closed container, amount of solution taken, given that, temperature is kept constant? Or will it change, if some extra gas is added to the container at constant temperature?
The equilibrium vapor pressure of a compound at a given temperature is an intrinsic property of the compound, meaning it does not depend on variables like volume of the container, quantity of the compound, etc. Other examples of intrinsic properties include density and melting point. None of these intrinsic properties depends on the quantity of the material or it's physical surroundings.
The simple answer is that in a thermodynamically closed system composed of a condensed phase (either solid or liquid) and a vapor phase of a chemical compound (or element), then only temperature matters - at equilibrium. This means vapor pressure may not be the same as partial pressure - even at equilibrium - even if the temperature is the same. That is; despite Dalton's Law of Partial Pressures, the partial pressure of a compound does depend on the other components present in the gas phase. This effect is usually so slight, when dealing with pressures around STP, that it can be ignored for a good approximation and Dalton's Law can be assumed to hold. You ask if the amount of material matters. Well, only if it isn't enough to provide for any condensed phase to exist after equilibrium has been established. That is, yes a drop of water in a volume of a cubic meter at 150°C will only exist in the vapor phase, and it's pressure will not correspond to the vapor pressure of steam at 150°C. The partial pressure of a compound depends on temperature, gravitational gradient, surface curvature, and the interactions of the compound with the other materials present; but it is a good approximation to assume it depends only on temperature (assuming enough of the compound is present to have some of it in a condensed phase). There's also another complication, which is surface absorption (onto the walls of the container), but again, this can generally be ignored.
YES, the vapor pressure will change if gas is added to a closed system.
Vapor Pressure is Temperature and Pressure dependent. So adding gas to a closed system increases the pressure, in turn changing vapor pressure.