How can you tell if Windows XP is 64bit or 32bit if you only have the partition/filesystem on a hard drive?

64-bit Windows XP is actually quite unlikely, 64-bit and larger memory configurations gained traction in the consumer markets sometime after XP, and XP was somewhat quirky due to the lack of manufacturer support for 64-bit.

That said the presence of two folders will tell you the system is 64-bit. If you have

C:\Windows\SysWOW64
C:\Program Files (x86)

Then your system is 64-bit. If both of those are missing then you have a 32-bit system.

Read more at The 'Program Files (x86)' and 'SysWOW64' folders explained

There will be 2 "program" folders if it is 64, That is 32bit in the image.

64bit XP was super rare due to hardly anyone wrote 64bit drivers for their hardware.

Registry

While the specific question of 64 vs 32 bit Windows XP can be most easily solved by looking at the folder structure, the generic issue of determining the installed Windows version (including the service pack level and specific build) from looking at the file system is properly solved by looking at the registry.

You can locate the registry hive files, and then look for the keys representing the Windows version; for recent versions of Windows they should be at HKLM\SOFTWARE\Microsoft\Windows NT\CurrentVersion and HKLM\SOFTWARE\Microsoft\Windows NT\CurrentVersion\BuildLabEx.