Not from that site. Eventually, if past practice indicates futurue performance, Microsoft will establish a corporate download page for XP. It exists for 98/ME. There you can download and save the patch or update and apply them to several of your computers as would be the case in a corporate environment. So far, I have not found such a site for XP. But, I'm sure it will come.
When it does, it is always about two months or so behind what is available on the Update site.
What that site does is download a file in a \TEMP folder. When it installs, it installs from that folder, and it is the reboot process that eliminates it. I have know people to halt that reboot and go search the \TEMP folders for the right size file . . . rename it and save it elsewhere and thus be able to use it later.