I've run a system like this, and yes...Air in the fluid (bottom of the bucket) should be on all the time. I ran mine with two air pumps, one for the airation of the
nutrients, and one for the bubbler/feeder tube. The feeder tube bubbler I only ran during hours of light.
This system is a mix between drip watering and Deep Water culture.
They work GREAT, Everything I grew in this type of system did GREAT!