My wife has been using Windows Vista on her computer, and it has finally borked bad enough that she is willing to try Ubuntu. The system is pretty much unbootable as far as I can tell, and I need to download and burn an ISO image of the 64-bit version of Oneiric Ocelot (11.10) for her. Can I do this from my 32-bit machine?
The bit version of the CD or DVD does not matter where you burn it. You can even burn a PC version of the 32 Bit or 64 Bit on a Mac. Same for windows and same if you are burning a 32 bit on a 64 Bit PC or vice versa.
What does matter is if you want to run it afterwards. Is it recommended to download the 64 Bit version if you have a lot of ram and the whole architecture is 64 Bit ONLY. This changes if it is one of those 32 Bit and 64 Bit architectures that support both.
It is also not necessary to run a 64 Bit on a PC that supports both, 32 Bit and 64 Bit but has 4GB or less of ram. The performance will not be any better if you use the 64 Bit. Stick with the 32 Bit for more compatibility in that case.
So in any case, don't worry about where you burn what type of CD/DVD.
Of course you can. The ISO image is a file.
The difference between 32-bit and 64-bit is a reference to the size of each instruction the computer processor can process. The wider the word or instruction the greater the information that the CPU can process at each clock tick.
By the way, there is backwards compatibility at the moment. A 32-bit operating system will run on a 64-bit CPU. I have done it. On the other hand a 64-bit OS will not run on a 32-bit CPU. The processor could not read the instructions. They are twice as wide as the instructions for a 32-bit CPU.
A CD will play in a DVD drive, but a DVD will not play in a CD drive. That is what could cause a problem, but I do not think that you have to worry about it.
It does not matter whether the OS burning the CD image is 32-bit or 64-bit. Any more than it would matter if you were wanting to copy music tracks from one machine to another. Likewise, they are files.
Burn the ISO image to disk, and put it in your wife's machine and boot. See what happens.
The same is true of Live USBs, by the way. You can use the ISO file of the 64bit installer on a 32 bit machine to make a 64 bit live usb. So if said computer can boot from a flash drive, the greener/cheeper option would be to turn a USB flash into a startup disk for a short while, either with Ubuntu's own Startup Disk Creator, or with Unetbootin.