No, you have nothing to worry about. Look at it this way:
APC makes the model numbers of most of their UPS's what the wattage rating is when drawn upon during the battery backup point.
An APC 350 is about a 350 watt UPS, capable of powering that much when it runs on the battery backup.
Volts X Amps = Watts
Your PC runs on DC power, at 5 to 12 volts, depending on the hardware it is powering.
Look at the PSU, it will tell you what the input and output voltage / amperage is.
Even though you have a 380 watt PSU, it will never use that much power. You might use 225 - 250 with your current hardware setup.
The thing that is very misleading about PSU wattage ratings is that they give that rating under an "ideal" condition. They can guarantee that rating at zero degress celcius, when the PSU is most efficiant. Once you put that PSU under load, inside your case running for hours on end, it is well above room temperature, and capable of putting out around 75% of the rated wattage.
Also, a PSU is only going to deliver as much power as required by the components plugged into it.
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