There is no udpate for flash player in Sony BDP-S390. If the icon for installing Adobe flash is selected when using the web browser, the message not supported on the device will appear on the screen. Use a computer or other Internet device that supports flash content to view these pages.
Yes, there is no update for Flash player installed in Browser of the Blu-ray Player as Charlie mentioned as it cannot be upgraded manually. It can be done only through firmware upgrades released by Sony.
You can check for system updates under "System Updates / Firmware" section of the below page.