The USB2Xchange isn't supported by the standard Ubuntu kernel.
Someone did develop a patch to support the Adaptec USBXchange and USB2Xchange; see http://linux.derkeiler.com/Mailing-Lists/Kernel/2005-09/4102.html and http://dl.exactcode.de/adaptec-usbxchange/
That patch is for quite an old kernel version though, and you'd have to apply it to your kernel version and compile it. That's not something a non-expert would be comfortable with.
As already suggested, probably the cheapest/fastest option would be to acquire a SCSI card for a desktop PC. Modern motherboards tend not to have PCI slots though, and PCI Express SCSI cards are uncommon (but not too expensive used).
If you're only able to connect via USB, you could try to find an older USB 1.1 USB-SCSI cable based on the Shuttle Technology chipset. Those do work in Linux. Some examples: Adaptec USBConnect 2000, Belkin F5U015-TPW, Belkin F5U115-UNV, Castlewood ORB USB Smart Cable, Entrega/PortGear USB-SCSI, Microtech XpressSCSI, Newer Technology uSCSI. (Sadly the asking price for those on eBay is quite high.)
USB 1.1 transfer rate is of course very low, but if it's a one-time transfer that shouldn't be too bad.