I'm using ubuntu 12.10 on intel nuc, and have no keyboard only a mouse attached.
Well, I took my time just for the fun of it to check all NUC Bios and Driver updates. It just so happen they use the same motherboard and offer the same BIOS Updates.
You can see the webpage where they all send me HERE
Since you do not have a keyboard, the option to press F7 when the system boots to read a pen drive and the Bios file in it is out of the question. This way only works when you boot the computer, after pressing the corresponding key it will read any pen drive you have inserted and look in the root folder of the pen drive for a BIOS file format for Intel.
Your only way is to Burn a CD with the BIOS update and boot from it. I will be assuming the computer tries to boot from a CD/DVD or a Flash Drive before going for the Hard drive. If this is not the case and it boots directly to your hard drive then you really need to get a keyboard since you are out of options.
You will need to get the GKPPT10H.86A.0036.BI.ZIP but for future reference, try to get either the file that ends in .ISO, .BIO or the one that ends in .ZIP. If you had the ISO file you just burn it to a CD/DVD and boot from it. In this particular case there is no ISO option.
The ZIP one, you decompress but here is the problem, the files need to be stored in a bootable CD/DVD or USB Flash drive in order to run (Again, assuming the BIOS reads this devices before the HDD). So you would need Windows to make it bootable as stated by the Intel Bios Update Instructions for this motherboard. After making it bootable you can proceed to copy the files to it and then later boot from it. The whole update process will be automatic.
Now that we have seen 2 options which depend slightly on having a keyboard or having the system boot into other devices before going to the HDD, we can now fall into the Bios Recovery option.
The Bios Recovery option found in the Bios Updates is just in case your BIOS fails to update but it is also used to update your Bios to that specific version, in this case, it will be updated to the latest. More information HERE
The part that you need to know is:
By default, this motherboard will first read any other device before going for the HDD. So if you have not yet changed anything in the boot process in the BIOS, you don't need to worry.
The BIOS-Update-Tools doesen't run under a OS like Ubuntu, it runs in a console bevor the OS is started. In the console you have no support for a mouse.
I would say it is impossible.