On my xp SP3 machine I wish to install the latest TOR update. I am not interested in Windows upgrade. Only TOR upgrade. Since I cannot use the installer package, how can I continue to upgrade manually?
You can attempt to install the 32-bit version of Tor Browser. If this fails, then there is no other native option with Windows XP.
windows xp is absolutely EOL and highly insecure!!! - current firefox/torbrowser won't work on this outdated system anymore.
not to mention that using outdated firefox/torbrowser is a bad idea and must also be considered insecure! - you probably would be even more secure (perhaps except in the view of anonymity and privacy) by using a current "normal" browser.
i also wouldn't advise you to upgrade windows, since current windows tracks you by default and it is much work resp. impossible to really fix this.
i highly recommend to use a FOSS-OS to run Tor on. probably Tails may also be interesting for you.