> How much energy will you need?
A lot.
Air compression is not cheap, energetically. Chilling is also energetically expensive. You'll need to do both. This would be pretty much the "brute force" method, like O(n!) in software engineering. The concept would be to bring half the planet's atmosphere down to near its liquid temperature. Currently in most US homes, Air Conditioning (air cooling) contributes the majority of each home's energy use. This would be much greater than that, it requires chilling air to -140 °C to get CO2 to "drop out".
In contrast, seeding the ocean with iron would be fairly close to O(1). However, potential side effects of this process concern some scientists greatly. Googling 'Russ George' may provide more insight.
Side note: The press material for sesinnovation also says the outlet stream is nearly 100% nitrogen...that means they're using a stream which already has oxygen removed from it (likely because the air was just used for combusting fuels). Air is roughly 21% oxygen so you can't take air, only remove CO2, and get 100% nitrogen out. Generally these "Carbon Capture" technologies only work on feed streams with high CO2. So hey're placed at industrial outlets, where fuel/coal is being burned. These streams can be > 10% CO2. Often they don't work as well, or at all, on diffuse streams like 0.04% atmospheric CO2. The technical reasons why this is the case deal with "partial pressures" as well as general entropy.