I have a 850watt PSU, and the only all in one UPS i could find compatible was the APC 2200va as previously mentioned here. But it doesn’t have much runtime (Runtime Graph), this is the cheapest option to buy. Anything DIY (Inverter/UPS + Batteries 24v) will cost you upwards of 25k depending on your retailer, but the runtime will be much higher because you can get higher capacity batteries.
It should as long as battery hasn’t gotten too old. 1600VA is 1600*0.6=960W & any typical ups battery deteriorates with time (from my experience, 1-2 years from battery mfg date which can be different from ups mfg date).