I'd have thought it would go OK, long as you give it a decent amount of RAM.
Let's see .... 2011 >> 2001 .... Thunderbird 1200C, 30GB hard drive, maybe an Athlon XP 1700 and a 7200 RPM 60GB drive if you were feeling extravagent. Yeah, I'd surf the web happily enough with that rig. Wouldn't be all that much slower than any laptop made today. But .... and it's a big but ... you'd need a lot more RAM than that system originally shipped with. 2GB and it would scoot along very nicely.
Merc is right, of course, but his ten year figure is a little out. Close though. Drooling for 1GHz and 256MB, that was more a 2000, early 2001 thing. After March 2000 you could just buy that 1GHz machine if you wanted it - they cost a lot at first, but the 900MHz Athlons were very nearly as quick and half the price.
Net result: off the top of his head with a casual comment, Merc is at worst about six months off the exact date - I reckon that's pretty good for an old fellah!