Did your home inspector recommend a testing company or abatement company? They are often a good source of info. Your agent and/or RE lawyer could be, too. Does the seller have to do a disclousure form where you are?
If you're an outgoing person, you could knock on doors in the building, say you're looking to buy unit XYZ, and ask about others' experience doing the same thing. Neighbors may also know if their unit has/had asbestos. Homeowners' association, maybe?