I recently purchased a Lightroom 5 software from Ebay but it is for a 64 bit computer and I have a 32 bit. I had installed the demo version on this laptop and it worked fine, is there any way I can download the 32 bit software demo and activate it with my 64 bit code on case?
Which operating system?
The Windows installer comes with 32 and 64 bit installers as standard.
If the demo version worked fine, then I don't see the issue at all, you have installed the proper version... Your serial number works for both 32bit and 64bit installations. All you need to do is apply your serial number.
By the way, I don't know what you bought on E-bay, but there is no such thing as purchasing only the 64bit version (at least, from honest salespeople there is no such thing)