How to keep house from getting so dusty with no HVAC system?
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.