I'm pretty sure that Echo has an internal clock so that it will still work when there's no internet connection.
I can't test it right now, but if anyone wants to try it let us know your results.
For the test, the Echo must be connected to the internet when you set the alarm.
e.g. "Alexa, set the alarm for 6:30pm"
Turn off or unplug your router/modem.
Wait till 6:30pm (or the time you set for the alarm) and see if the alarm sounds.
Cheers!