Probably the most common mistake is being too anxious to get tenants. Getting the wrong tenant is worse than no tenant at all. The solution is to check the employment, credit and criminal history of each new tenant. Instead of the apartment owner doing this, consider using a professional screening service. Should a legal dispute arise, these documents will greatly help the owner in court.
Evictions are an unpleasant reality.
Some states can grant an eviction in 24 hours, but in most places it takes weeks or months. In the meantime a problem tenant can occupy the apartment until the eviction judgement is rendered. Another good resource is the Institute for Real Estate Management. Older, historic buildings may look like they present a great opportunity for rehabbing, but first bring in a building engineer to scout out the problem areas. In many cases, the older the building, the more potential problems.