Never really used the alarm system, so I tested it a few times. A few things I've noticed:

-Reminder (Minutes) uses normal minutes, even if you have Eorzea Time selected as Clock Type.
-If you have a Reminder set, that's the only time when the Alarm will sound. Without a Reminder set, the Alarm will sound at the exact time you've set.
-The Alarm is part of System Sounds, so make sure to not have that one disabled in the system config.