I've had a central HVAC system and always used high quality furance filters all my life until my current apartment, which has basically no air filtration/circulation system. (Radiator heat, and window ac units) It seems to get dusty extremely quickly, much quicker than any other place I've lived with a central system. Having a dog doesn't help things either. I purchased an expensive standalone air purifier, but that really doesn't seem to be cutting down on the dust much either. Any suggestions?
I'd start by trying to figure out where the dust is coming from. Do you now have carpets where you used to have hardwood or other non-fiber flooring surfaces? Especially old or broken-down carpets can be a big source of dust. Also you could have a different environment outside that has more dust in it compared to your previous location(s) that's working its way into your apartment? Possibly a different cooking environment (or no vent hood to evacuate cooking odors). If you do have a vent hood, beware that with many of the vent hoods they don't actually remove the air, they just run it through a charcoal grease filter and send it right back out into the air.
It's also possible that your old places with the high-quality HVAC filters would be similarly dusty if you hadn't had those filters in place.
Probably need to figure out where the dust is coming from, and try to mitigate that as much as possible. If that's not possible, then probably the only other solution is to buy multiple air purifier units and distribute them throughout the rooms of your apartment. If the noise is objectionable while you're in your apartment, you could put them on timers to run only when you're not there.
That is a hepa and carbon filter unit. Had to look it up to figure out how it took care of the dog smell. Our current house has cable heat . No central air and the dust and dog hair collect at an amazing rate. We have wood / vinyl floors so a big vacuum would be a waste. I looked at smaller battery powered cyclone type of vacuums but after reading all the reviews ended up purchasing a "Shark rocket" it works great to pick up the hair and dust on the floor. We have had it less than a year but so far it has at least help pick up the dust bunnies and hair balls. It also works on our area rugs all very low pile, not sure if it would work with longer shag carpet. Two long for coment so moving to an answer. This has helped reduce the visible dust in our small house.