Should run okayish with 2000 tbh, but may not like SP4 if you have updated it due to increased RAM usage - if you havent already, it would be worth changing any non-essential Windows services to "manual" or even "diasbled". A good browser for a low spec system like this would be K-Meleon - check it out, only uses about 15 - 20MB out of the box, and then a further ~2MB per tab. It uses the Mozilla source (ala Firefox and Opera) and is highly customizable, as well as fully compliant with everything Firefox, and supports most plugins.
Another light browser, Google Chrome wont work in 2000, but there is an alternative built on the same source code that works in it called Iron, by SRWare.
I use 2000 Pro as my backup OS on my old desktop and its still as solid as a rock, and fast as fook (running it on a Pentium III Tualatin 1.4Ghz with 512MB RAM) - its a business OS, so it has to be efficient, secure and stable.