|The BIOS doesn't care about file systems, so updating it won't change anything with regards to the file system.|
If I'm reading what you're trying to do correctly, you're trying to boot the hard drive from one computer in another. This will usually not work with Windows 2000 and newer, as they typically use specific chipset drivers to access the hard drive. You would have to boot from the Windows 2000 install disc and "repair" the installation to get it to boot in a different computer.
As for backing up the data, there are many tools out there that can read NTFS file systems from Windows 9x/Me, if that's all you want to do.