Is it possible to install flash firmware update without having access to OSX?
Don't know if this is directly possible. Maybe via the repair partition (press and hold CMD+r after the startup chime) of Os X - assuming that you do not have deleted it to and used Mountain Lion or Lion prior on the Macbook.

What should work but take some time to do so is to install your old operating system and  either than download the flash update and install it and install a „clean“ install of Mavericks after this procedure or just install Mavericks right on top of it via the Mac Appstore.