The answer is, possibly.
The architecture of your computer is fixed, and is evidently 64-bit since you've had a 64-bit copy of Windows 8 installed on it previously. However, while it's not possible for a 32-bit computer to run 64-bit software, it's entirely possible to install a 32-bit version of Windows onto a 64-bit machine.
Check the copy of Windows you've got. Not all of them say, but depending on your disc it might be possible to install a 64-bit version instead with the same DVD - there will be a "version choice" menu if this is the case.
A quick way to check if your current installation is 64-bit is to examine your C:\ drive - look for the Program Files (x86) folder. Double-check by looking at the process list in the task manager - in 64-bit, running 32-bit processes are marked with a *32.
If neither of these markers are present, you're running 32-bit, and to use more than 4GB of RAM you'll need to reinstall.
Edit: The implications, added later, are essentially this:
- 32 bit Windows takes slightly less space and memory to run.
- 32 bit Windows cannot, as you've noticed, run 64-bit programs
- 32 bit Windows cannot address 4GiB of RAM - if your computer has more, Windows won't make use of it.
- 64 bit Windows cannot run legacy 16-bit Windows applications, like older DOS applications
There are some more, but I think those are key. It's not the end of the world, especially if you've not passed the RAM limit - and remember that mostly, programs and games aren't 64-bit anyway, and won't be able to address 4GiB of RAM even on a 64-bit OS.