Ah, well it is showing only "there are currently no drivers for this system" when I first go there. Visiting the website a second time suddenly shows content.
You may explore your BIOS if there is any option to load a BIOS update from a USB thumb drive. I'm not big on laptops, but I always by motherboards for my desktop that support this feature. It may be common enough by now to be supported on your laptop.
Alternatively, if you open the BIOS 1.24 zip file and go into the BIOS_Acer_1.24_Windows directory, you'll find it has a DOS and a winflash directory. The DOS directory contains the files to flash your BIOS from DOS. There is a project called FreeDOS, which many people have used to create a bootable DOS compatible disk image from which they then boot and flash the BIOS. I'm not sure what guide is best for that, as I've only found guides to make a bootable floppy and that won't be big enough to fit the BIOS flash files.
Or you could burn the DOS directory from the BIOS 1.24 zip file to a CD, and then make a bootable DOS floppy image that you put on a USB thumb drive. Then boot from the USB thumb drive, and start the BIOS flash from the CD. Perhaps here is something helpful for that: http://www.bootdisk.com/