Hello,
It seems to me that if you have a small tube in the lower part of the bottle and the upper part of the bottle is completely closed, nothing will flow out the lower tube. The upper part of the bottle would have to have an air vent to allow air to enter the bottle from the top.
There is a problem in doing it this way however and that is that a container with a hole in the bottom does not meter out water at the same rate all the time if the container is being emptied. That's because the pressure changes as the liquid level decreases. That means it may be hard to meter out the correct amount.
To make up for this error you can note that it is very systematic and thus predictable. You'd only have to do a few tests to determine how to vary the 'open' time period having the previous history stored in memory, and have the system reset every time the bottle is refilled, however you'd have to refill it to the same level each time. That assumes you dont want to be bothered with measuring the liquid level either.
Also, by opening the upper tube to allow air to enter the bottle that's when the flow would start not using a solenoid valve unless the lower tube diameter was large. That means it might even be possible to use a very small solenoid air valve rather than a valve that has to come into contact with the liquid constantly.