Yes, this can work... if the aeroplankton is thick enough and nutritious enough, and the creature has some way of absorbing sufficient water. Living near waterfalls or in a humid environment would probably suffice. For aeroplankton to exist, high humidity is also likely.
As the OP says, this would require lungs that serve two purposes, those being gas exchange and nutrient collection.
I think that most people would be sufficiently familiar with the respiratory function of lungs that I don't need to go into too many details. That just leaves the matter of nutrient collection.
To understand how this creature would feed, we must first understand what it feeds upon. Aeroplankton would seem to be microscopic organisms light enough to float in the air and live there, being either phytoaeroplankton photosynthesising for their living, or zooaeroplankton feeding upon the floating microscopic plants. Either way, these microscopic organisms would need to be small, lightly built, and have a large surface area in order to be able to float in the air on the air currents. This would make them quite flimsy - even the plants - so they wouldn't be as tough as organisms that rely upon their exteriors to protect them at the expense of increased weight or density. They would likely reproduce very rapidly in order to compensate for the likelihood that many will die from the many hazards in their environment.
So, we have either single-celled or tiny multicellular organisms that float in the air. To be able to survive, they must exist within a certain size range dictated by the air pressure and gravity, not too heavy to float, yet not too light to sacrifice efficiency. That they will occur in a limited range of sizes (whatever that range may be in their world's conditions) is an important factor in this animal's strategy.
It may not be known to many people that aerosol medication can be targeted to affect a particular part of a patient's respiratory system... but what does this have to do with this filter-feeding creature? The mechanism of this selectivity is the droplet size of the aerosol. The smaller the droplets, the further into the respiratory system they will travel before colliding with the wall of the respiratory system.
So, knowing the size of the aeroplankton, we can work out how far into the creature's lungs they will travel, and the size of the air passages at which a collision will most likely occur, and likewise, the creature can evolve to maximise the surface area of its breathing passages at the point and passage size at which the aeroplankton will most likely impact.
Higher up in the lungs, where the passages are larger, and only a few aeroplankton are likely to impact the walls, the walls may be dry and not sticky, to allow the aeroplankton to bounce off and continue further into the lungs.
At the point in the creature's lungs where most aeroplankton will impact, the walls would be coated with a layer of mucus designed to trap any aeroplankton that touches it. Beneath that mucus would be cells evolved to devour the trapped aeroplankton and digest them, along with cells to produce the mucus.
Beyond the point at which the aeroplankton is trapped and digested, the narrowing passages would carry air - now free of aeroplankton - to the respiratory part of the lungs, where gas exchange would occur.
So... one "organ", the "lungs" would serve the purpose of both respiration and digestion.
As to how this might evolve, consider the consequences of not filtering out the aeroplankton. Any creature which lives in an aeroplankton-rich environment would risk respiratory infections from a failure to filter out the tiny organisms. Some of those creatures which could perform such filtration may have done so by digesting the trapped aeroplankton in-situ. With sufficient aeroplankton trapped, the creature's metabolic requirements may have been partially satisfied, and as they became able to trap more aeroplankton, its metabolic requirements would have been fully satisfied, leading to the degeneration of the normal, unnecessary digestive organs.
I should point out that such a lifestyle would involve mainly sitting still and breathing, and would suit a low-metabolism, cold-blooded creature. It may well be camouflaged and/or contain toxins to hide from - and/or discourage -predators. Having to move to escape a predator would take a lot of energy that would take a long time to replace given its feeding strategy.