As I mentioned in chat, this project isn't possible as is.
As with an engine in a vehicle, computer processors can very dramatically. A boat, plane, train, and car all have an engine, designed to move the vehicle from point A to B. Unfortunately, that's more or less where the similarities end. You can't take the engine out of the the boat, and put it in your car. They're just not compatible.
Likewise, processors have fundamental differences known as architectures. Your 32-bit PC is running an
x86 processor, while the Raspberry Pi (and most phones and tablets) are running an
ARM processor. As with the example above, the architecture is fundamentally different, and not swappable.
Theoretically, you could swap out components, and change the appropriate configurations, but it defeats the purpose of what you're trying to do. The kernel isn't the only thing that would need to be swapped. You need to recompile literally every single binary on the system, including the software you're trying not to lose.
If you can't download the application anymore, I doubt you have access to the source code, which means recompiling it isn't an option.
You could probably clone the old computer, and run it as a VM in a newer computer. It doesn't even need to be a dedicated computer. (See: VirtualBox)
You might be able to use an x86 development board if you really want to stick with a setup of this style. Minnowboards are significantly more expensive, but run on the correct architecture. I believe you could get an image of your current system running on it.