It seems to me this would work only if the cup to which you transfer water is lower than the source cup. The experiment you describe would not work.
Capillary action is the result of adhesion, cohesion, and surface tension. If water molecules are attracted via intermolecular bonds to another material with which they have an affinity even greater than the cohesion they have with each other, they will creep up the surface of the other material, even against gravitational force. If the surface of the other material is porous, the water can climb quite a distance.
Other water molecules will be pulled along because of cohesion to the creeping water, and surface tension will hold the creeping water together. But when the weight of the climbing water overcomes adhesion and/or cohesion, the column of water will stop climbing.
You can route a rope from one cup to another, but in order for the water to drop off the rope into the destination cup, the weight of water in the rope over that cup would have to overcome adhesive and cohesive forces that got the water there in the first place. If the destination cup is higher than the source cup, you would need a height of rope that would stop water from climbing out of the source cup. Any lesser height, and water will never leave the rope over the destination cup, because the weight of water in the rope will be insufficient to overcome adhesion and cohesion. In order for water to drop off, the height of the source portion of the rope necessarily would be greater than allowable for capillary transport against gravity.