Can a liquid exert pressure upwards ?
Yes water molecules being free to move in any direction exert pressure in all directions so they exert pressure upwards also. But the magnitude of this pressure differs when compared to the water pressure felt but at the base of the container. This difference arises primarily due to the length of the water column which compounds with that of the water creating an additional pressure.
This does not mean that due to the upward pressure water moves upward from a container into air. This does not happen because of force of gravity which pulls water molecules in an antipodal direction.
If however, sufficient heat energy is applied thereby infusing Kinetic energy to the same water molecules , this extra energy helps water molecules to move upward overcoming the force of gravity resulting in evaporation of water.