There should be no problems with this. This should work on XP, Vista and Linux systems. I have a usb cdrom that has worked on my xp and now works on my cousins Visa. She should be fine. One thing you can do is google 'model review' replace model with the brand/make and read up.
Good luck, but she shouldn't need it.
Frosty how did you check to see if the CDROM is not working. To me that link you provided is more than enough for her. Just to read CD-ROMs. The internal though would be better in my opinion. There is brand names for cheaper than the external and the life expentancy* should be much longer than this non-name brand CD drive?
(off topic - when you said old lady I though you referred to the computer lols).
Just to add to what others have said about getting an internal drive (I agree btw) one other main advantage of going this route is that you should never run into a problem if the OS needs to be reinstalled. If that is an older computer you should consider this as it might not be able to boot from USB (and hence the CD drive). Check this out in the BIOS first before you go one way or another.