hmmm... well what I would suggest is getting one of the cup dispenser things that they usually have for bathrooms except those are small cups. Get one of those and prop it upside down, then have a suction cup on a pneumatic arm go up, grab the cup on the inside bottom, bring it down below the cup dispenser, have a second suction arm grab it, first one lets go and drops down out of the cup, second one brings it under the fluid nozzle and rotates it 180? so its facing upwards...
Or maybe have the cups sitting in a cylinder right side up and the inner diameter of the cylinder would be slightly larger than the diameter of the cups. At the entrance of the cylinder, there would be 2 iris's. One iris would be holding the first cup from falling out at random by holding it by the lip(the top outer rim), the 2nd iris would be holding the second cup from falling out when the first cup is dispensed also by holding it by the lip. You send a command to the first iris to open, the first cup falls down into a track or something, the first iris then closes when the first cup falls out, the second iris then opens dropping the stack up cups down by one, and then closes to hold the 3rd cup from falling when the second cup is dispensed. and so on and so forth
Hope these ideas helped you in any way