a new low budget PC instead?
Yes. The current processor is far too slow, and the motherboard will almost certainly not be capable of supporting the kind of upgrade you will need.
Money spent on upgrading such a machine to a broadband - friendly state would be better put towards a new, faster machine. A modern budget machine will be more than capable of doing the job, and needn't cost a fortune. It will also come with Windows XP, which makes internet connection sharing/networking a breeze.