To answer your question yes you can upgrade an old Intel® Pentium® III processor to any processor within that family and socket http://ark.intel.com/ProductCollection.aspx?familyID=590&MarketSegment=DT. The real problem is going to be finding a processor that would be an upgrade (just so you know they are all been discontinued for close to 9 years now I think).
Comparatively speaking even the low end new computer today would be a great deal more powerful than anything that you could upgrade to within the Intel Pentium III line.
Christian Wood
Intel Enthusiast Team