In order to update the BIOS you would need an external USB CD or floppy drive. I believe that the original floppy drive with this model utilised the PC card slot. However there is only a roll back BIOS available for your model:
“This BIOS recovery update is applicable if you have already updated your VAIO using the Windows XP BIOS and want now to reinstall the original operating system (Windows 2000) on your VAIO.”
Assuming that you still have the original operating system then the shutdown problem is unlikely to be connected with BIOS as, if it was, this problem would hopefully have been addressed by Sony years ago.
I suggest that you download a temperature monitoring utility such as PC-Wizard and let us know the CPU temperature.
You say that you have added a new big fan over the CPU. Is this externally or have you had a new fan fitted?