Wondering if my pre-built can run this GPU?

Recommended power supply for a 960 is 400w so your PC should be able to handle it. However due to the nature of pre-built PCs usually having subpar PSUs installed in them, I think some caution is necessary when upgrading.
 
I wouldn't risk putting a GTX 960 on a generic 460w PSU. Lower end PSUs sometimes take other components with it when dying (worst case scenario) and they usually cannot output the wattage that is written on them. It's also entering 4th year of its life, so I can't stress enough how important it is to get a new one ASAP.