If the possession charge resulted in a nolle prosequi, then that means that the prosecutor ultimately dropped the charges and the case should have been dismissed. If the case was dismissed then this should not result in a criminal conviction. The problem with most of these third party background check companies is that they only report that you were charged with the crime, but do not accurately reflect that the case was dismissed. If you have a copy of your dismissal, you should show that to the apartment mgmt.