Don't use blinds to cover up your windows. Anybody who has ever owned these knows what kind of dust magnet they truly are, especially in a small apartment. Instead, use small curtains made up of a light fabric.
Place small rugs in your apartment and then clean them once a week. These rugs will actually collect quite a bit of dust. However, since they're small, you can easily remove them from your apartment whenever you feel the need to do so. Simply taking these rugs outside and beating them or even shaking them will remove dust.
Keep your windows closed whenever possible. Open windows allow outside particles to enter your small apartment, creating more dust in your home. Place a screen in front of your windows for when they are open. Be sure to clean those screens often in order to reduce dust in your small apartment.
Keep clothes in dressers or closets. Clothes attract dust and can even create dust due to pieces of cloth and skin particles. Avoid putting your clothes on the floor or other places.
Wash your bedding often. The covers on your bed may create and gather the most dust in your small apartment. Washing your sheets will remove the flakes of skin and loose fibers found on the sheets.
Use a damp paper towel or washcloth to eliminate dust. Using a broom or other duster spreads dust throughout the air and surfaces. Instead, gather the dust on these towels and dispose of them. If you use a washcloth, throughly rinse it when you've finished gathering dust.