Good question. In fact, there are many different ways that maintenance requests can be handled. Some newer buildings have an online system for maintenance requests that you can access day or night. In some buildings that have a doorman, usually calling or asking the doorman is enough to have the maintenance request handled. Lastly, in smaller buildings that don't have a doorman, you will typically get the phone number of the superintendent when you move in. This is the person to whom you will direct your maintenance requests.
That being said, even when you have successfully submitted a request for maintenance, the speed and efficiency with which the request is handled depends completely on the building/management company. A seasoned agent should be able to tell you what sort of reputation a building/management company has for processing maintenance requests as you hunt for a new home.
I hope this answer your question!