Someone on the Asus board is misinformed. PCI Express gear is all (in theory at least) backward-compatible. That is, a PCI Express 2.0 device will work in an older PCI Express 1.1 or 1.0 slot, but at reduced speed.
Boilerplate like "Supports PCI Express Base Specification Revision 2.0" is just bragging that the card can, in theory, ship data at the speed of a PCI Express 2.0 slot, not that it is incompatible with the older slot type.
In reality, you will not come close to the limits of PCI Express 1.0. An external hard drive will hardly push 1.0 Gbit/s on its best day, and PCI Express 1.0 can handle 2.5 Gbit/s.
Do you have another PCI Express slot you could try? A 1x card should still work in a larger slot, for example 4x or 16x. It does on my motherboard.
Unfortunately, there is no way to "update" PCI Express 1.0 to 2.0. The electrical specs are different. You would only get PCI Express 2.0 with a new motherboard.
Your problem could just be one of those weird, rare cases of incompatibility in gear that's supposed to be compatible. It happens.