I have a Sony Cybershot DSC-P92 and it always has been working properly untill today. When I inserted a fresh pair of charged batteries the powerlight immediatly comes on, even before I turned the camera on. When I press the powerbutton and the camera starts I have to re-set the date, which is odd. After that done and ready for use the flash doesn't go off and it starts beeping and displays an error message: E:91.01 Is this problem fixable or should I throw it out of the window (metaforically speaking)

Haven't I wanted to Throw it out the window before as well. That was a good chuckle but no you don't need to do that, what's happened is that the internal battery the one that keep the day date time and anything else programed into the camera alive while the main power source is shut off.
Replacing an internal battery is covered in the owners and can be done most often by the user/owner. It looks like a watch battery but unlike a watch battery can be replaced without a whole lot of difficulty. I'm unable to find a manual file for your camera but would suggest because of design that the internal battery would be in the same compartment as your main power batteries and memory card it will be under a little door don't forget we are looking for something the size of a watch battery. Hope this helps. Oh and close the window ;-)