A cup will never be sufficient in terms of water volume, unless you want to be refilling it all day every day.
A large container (over 5 liters atleast) will serve you better. It must have strong (prefreably moulded/part of the container) handle/s of some kind, so it can be hung,
(unless you have a high flat surface to sit it on) above the cage, so that gravity assist the flow.
Simply inserting a foot or so of standard aquarium airline into the existing opening , perhaps through a hole in the lid, so it holds it in place, will do.
At a point along the airline thats convenient to you, between the container and the drinking end of the dripper (outside the cage) ,cut the line and insert the plastic flow valve (that I posted a pic of in the other thread), these are very cheap and can be bought along with the airline.
Then the remaining airline you insert into the cage by whatever means, so that the end
hangs or is held in place, easily accessable to your lizard from a favorite perch or basking spot.
If you like, you can insert some kind of small plastic gadget into the drinkingend of the airline, to slow the flow further, but this is not nessesary.
Now you must 'prime' your dripper (get the water flowing) by pretending the drinking end is a straw and sucking till the water flows.
Now you turn the flow valve till the flow slows to a drip rate you find acceptable.
Now if your container holds enough water, that it will not run out (or stop flowing due to low level of water), then at the end of the day, you turn the flow valve till the dripping stops altogether. This works as a fawcett, no need to be sucking on it again the next morning, just open the flow valve.
Ideally, your container should last one full day, and still not have its water level fall so low that dripping stops. Then you simply open its lid and refill it, still no need to Re-prime it.
Atleast once a week its a good idea to clean your container out (hot water no chemicals) You can add a little clean sand/salt/hot water and shake the hell out of it, sand/salt is abrasive) clean and refil, to avoid buildup of nasty bacteria.
Build a man a fire and he's warm for a day, set him on fire and he's warm for the rest of his life!