It looks like they also have a built in tool to install the BIOS upgrades. In the Manual also provided on the ASUS site page A-15.
- Download the BIOS upgrade from the link you provided.
Put the extracted file on a USB partition, using "FAT" file system not fat32 as stated here.
Plug in the USB, then turn on the computer, press the key to get into BIOS on boot.
From the Manual follow these directions:
- Use the Start Easy Flash function on the Advanced page of the BIOS Setup Utility. Follow the instructions shown.
- Locate the latest BIOS file and start updating (flashing) the BIOS.
- You must Restore Defaults on the Exit page after updating (flashing) the BIOS.
With this you should be able to upgrade your BIOS without Windows.