Is there any monitoring system for docker which "out of the box" supports sending alerts(email/smth else) in case of "Container#111 is down"? Or does implementation through the Nagios/Zabbix more right way?
Docker doesn't provide any "alerting" feature. It's task for your monitoring tool.
Zabbix world:
Zabbix supports email, SMS, ... out of the box + you have community scripts, which can be used as well: http://monitoringartist.github.io/zabbix-searcher/#script_alert
Docker monitoring solutions for Zabbix: http://monitoringartist.github.io/zabbix-searcher/#docker