Power up after UPS initiated shutdown
Simply put, no. The APC/Apple software is designed for shutdown only. I'd been looking for a similar solution for years when I managed a few XServes, but my institution opted to provide massive battery backups and a gas-powered emergency generator when we moved to a double rack of Windows servers a few years ago.

This device 3Gstore Remote Power IP Switch - 1 Outlet looks like it might work. I am going to get one of my very own for a security web-cam I just installed at home.

Disclaimer: Not professionally or materially connected with 3Gstore.