The way I read this, when you have a computer plugged into your UPS, if the power to the UPS is cut, the UPS beeps as though it is providing power, but the computer shuts off...? Is that right?
Does the monitor also lose power, or just the computer?
Check the outlets on your UPS; some may offer surge protection only; make sure the PC and monitor are both plugged into outlets that provide backup power as well as surge protection.
If the PC shuts off, but the monitor does not, are you using a modern, active PFC PSU in your computer? Many newer PSUs do not care for the simulated sine waves of typical UPS units, and will shut off. If that is the case, you'll need to get a better UPS. APC's Smart Power series produces a true sine wave, and Cyberpower now has some models out that, while not true sine, have a waveform that will work with modern PSUs.