This is a bit trickier than you might think as you need a FAT-16 formatted USB stick of less than 2GB, otherwise the Eee updater gives you error messages saying it cannot read 1000.ROM or that it is in the wrong format.
Anyway, to save time for anyone else who is searching how to do this, see http://wiki.eeeuser.com/howto:updatebios.
From Windows, you can right click a mounted USB and choose Format / FAT to get the correct FAT-16 format, then download the latest Eee 1000 BIOS .ZIP file from the Asus site (mine was 1000-ASUS-1003.zip), unzip it and rename the 1000-asus-1003.rom file to 1000.ROM before copying to the USB stick. Put it in your Eee then power on with ALT-F2 held down, making sure you are running on mains power and have a fully charged battery. It takes about 5 minutes to flash update the BIOS ROM.