Shutdown is pretty easy. Create a scheduled task with the following command line:
shutdown /s /t 60 /c "Scheduled shutdown"
That will shut down the computer after popping up a 60-second warning. It will force any stuck programs to close, and will leave a comment in the system event log.
I can't think of any feature of Windows that would let you schedule a startup - that's normally the domain of the BIOS. In a quick test I performed on a Dell workstation, I had to enable "Auto Power On", then set the power-on time. I suspect that some other BIOS's may be less intuitive.
For a more exotic and Rube Goldberg-esque solution, you could enable wake-on-LAN and send a wake-up packet from a neighboring computer. However, I've never been able to make that work myself.
Finally, I have to wonder whether you should even be turning off your server in the first place. The vast majority of servers stay on all the time.