So, is it possible to upgrade my operating system or is it cheaper to just get a whole new PC?
XP Home edition upgrade can be had for as little as around $75 (I just checked Amazon). You may want to put another RAM chip in first. You can turn off the eye candy and have it almost look like 98, and it'll run faster. I had XP running like this on an old 350 HMz laptop with 160 MB RAM (bought a 128 MB stick to extend the 32MB) and it ran fine for basic stuff like email and Web browsing.
Even cheaper of course is seeing if Linux will install. Get a CD of Knoppix or such, put it in the CD-ROM drive and boot from it (it won't touch your current Windows install). You know you're okay for a full install on the hard drive if everything works fine.