JellyBean Firmware 6.2.A.0.399 for the Xperia P LT22i, which is in testing stages has been leaked by DooMLoRD from xda-developers. The developer has managed to flash the firmware to the consumer device and has made it available for all.
However it's not recommended for you to install firmware as it is incomplete yet. The firmware is said to disable GSM support for your Xperia P, which means you won't be able to make calls. Also, DooMLoRD has posted a warning saying the test firmware is meant for only Xperia P and therefore should not cross-flashed with other devices.
It is not advisable for you to install the update unless you thoroughly know what you are doing. Anyways, DooMLoRD has posted instructions on how to install JellyBean test firmware.
Unfortunately the JellyBean update does not include small app. However, the camera app is said to be updated. Clear Audio+, Xloud, the equalizer and Clear Phase along with Stamina mode are included in the update. Also, The latest media apps are also present along with the new lock screen. Moreover, as per the leak the Kernal for will not be updated with the new version.
There is no information on when Sony might release the Android update officially for the Smartphone. But, it might contain more features then the test firmware, as the leaked update does not present the full build.