That is actually not the hardware requirement for the burner but for the software that comes with it.
I don't think you'll have a problem with it, but with such an old computer, which has not only a slower CPU but also a slow hard drive, you will probably not be able to burn DVD-Rs at the highest possible speeds (8-16x). Not a big deal really, it just takes more time to burn the disc.
A year ago, I still had an Athlon 1000 with 384MB of RAM and was able to burn DVDs at 4x speed, using Nero 6 (which is better than the overloaded Nero 7 in my view). With 8x speed, there were occasional problems, so I preferred the slower and safer speed.