Were it me, I'd tinker around to try to save a nickle, but in your case there are inexpensive PCI SCSI controllers available that would give you the speed you need to run that drive properly.
However, if you feel like tinkering, some considerations:
First, I would check to see if the terminator is Single Ended or Low Voltage Differential. The old card you mentioned is only Single Ended, and might not work if improperly terminated.
Second, you may have to set a jumper on the Tape drive to Force Single Ended operation.
Verify that the drive is not a wide drive, because if it is you will need special parts to set-up a wide device on a narrow controller.
Having verified those things FIRST. It is possible that it will work. Your next obstacle is the OS, you didn't mention what OS you are using. Windows NT and its derivatives W2k, W-XP, WS 2003 are all relatively SCSI friendly W95, W98 not so friendly. Your version of windows backup could very well work with this drive, as some of them were written by adaptec, optionally you can purchase one from Adaptec that would probably work. The drivers for that card are probably easy to come by, but SCSI being a multi-class interface, you will need a scsi class driver for the tape drive as well as the actual device driver. Since the CD class driver is aspicd, and the disk class driver is aspidisk, I suspect you will find that the tape class driver is aspitape. (If I was at home I could look on my own PC and tell you, unfortunately I'm on vacation).
BTW have you checked the cost of DLT Tapes? And don't neglect to get a DLT Cleaning Cartridge as well.