Are there any out of box API (webservice) exposed from Appian for external syste

If the task is performed by an external system, you should not have a user input task in Appian. Consider using a call to web service that can send the data to your external system and receive the response back (if its done immediately)
On the other hand, if you have a user input task that can be done both as a user input task in Appian as well as by an external system, you can expose an Appian process as a web service and use this WS from the external system to notify Appian of completion of task. You need an exception on your user input task based on Messaging to allow the task to be skipped if the job is completed externally first.

If the task is performed by an external system, you should not have a user input task in Appian. Consider using a call to web service that can send the data to your external system and receive the response back (if its done immediately)
On the other hand, if you have a user input task that can be done both as a user input task in Appian as well as by an external system, you can expose an Appian process as a web service and use this WS from the external system to notify Appian of completion of task. You need an exception on your user input task based on Messaging to allow the task to be skipped if the job is completed externally first.