Cygwin stopped being compatible with Windows XP in 2016. Following this post on stackoverflow, you can find the setup files and a (very slow) mirror that’s compatible.
I made my own, considerably faster, mirror of the last Windows XP compatible Cygwin release, at http://cygwinxp.cathedral-networks.org/
The setup file is available in the cathedral subdirectory, http://cygwinxp.cathedral-networks.org/cathedral/, along with a batch file that contains the command line options required to install from that mirror. I suggest you download both and run the batch file. If you only want the setup file, the essential option is the –no-verify one, which disables the use of package signatures. The signatures were not available in the Cygwin archive mirror, so I don’t have them.