Most APC UPS will cut power once they send the critical battery status and after a predefined delay.
Most UPSes (not only APC) will cut the power in similar case (or they'll simply die, not much of a difference). But consider this scenario: upon receiving battery critical (or even earlier if shutdown takes considerable time to finish) machine performs shutdown. But before UPS cuts the power, utility power returns. UPS will not cut the power towards machine and machine will not get started again. So it would actually be better if machine didn't perform shutdown. If machine has advanced control over UPS, then it can order UPS shutdown as the very last step of emergency shutdown and if that fails (because utility power returned) it just performs reboot.
But I highly doubt ROS supports that kind of interaction with UPS. If it doesn't, then battery critical event sent from UPS is useless (not only because ROS shutdown is a joke anyway). And if MT is not the only device connected to UPS, then it doesn't make any sense to connect control connection to it ... as most devices will make much better use of events triggered by UPS and are usually more sensitive to power loss. ROS device will likely survive sudden power loss just fine (much better than some NAS or similar). It will still benefit from other functions a typical UPS performs (brownout protection, delayed start protecting from current inrush, etc).