Hi,
In theory there is no reason why you shouldn't get the speed upgrade as long as you use the card you linked to and a compatible enclosure to connect to it. The USB3.0 card is connecting via the PCIE bus so there is plenty of bandwidth there. Make sure that you have the correct drivers installed otherwise you will be limited speedwise.
A BIOS update wouldn't provide the speed increase anyway as it would also require the hardware to be upgraded too.