"64-bit Installation Notes
Windows: The Lightroom installer contains both a 32-bit and 64-bit version of the application. By default, the 64-bit version will be installed only on Windows Vista and Windows 7 64-bit operating systems. All other operating systems will install the 32-bit version by default."
I Downloaded LR 4 online on a windows 7 64 bit system.
Download system doesn't matter.
I just purchased a new windows 8 system that is 32 bit.
If I were you, I'd take it back and get a 64-bit system, unless the hardware won't support more than ~3G ram anyway.
If you can't take it back, buy an OEM version of Windows 8 64-bits and install it yourself.
Not good to be constrained by a 32-bit system these days, especially if you'll be running Lightroom (or Photoshop, or ....... ).
And make sure you get at least 4GB of ram, preferrable 6GB or 8GB or more, if you can.
Will software backup work on new system?
I assume you mean "can you install the downloaded software on your Win8 machine" (not sure why you used the term "backup").
The answer is yes, but see notes above: I wouldn't. Your call obviously..
it actually was given to me trying to make sure it will work before I transfer my backup of software to new system. It does have 6 ram
It does have 6 ram
Won't matter if it's a 32 bit system. Only up to 3.2GB will be seen and used by Windows.
You should run 64-bit Windows on the new computer if the hardware (CPU) is compatible with a 64-bit OS.