I have done this once, on Debian; my experience ought to apply to Ubuntu too. Unlike you, I had a planned upgrade, so always had the option to go back to the old hardware if necessary.
Short answer: yes, you'll be able to boot, assuming that you have something like a stock kernel on the disk, and sufficient drivers built-in (or in an initrd) to be able to mount the disk. In practice, that means the appropriate SATA or SCSI drivers must be included.
Once you're up and running, you should be able to use
dpkg --add-architecture to tell your system you want
amd64 packages as well as i386 ones, and a subsequent
apt-get update (or
aptitude update) will offer both i386 and amd64 versions of packages. Install a 64-bit kernel and nothing else.
Reboot into the new kernel. Verify (using
uname -m) that you're actually running the new kernel. If it boots but fails to start userland, it may well be that you didn't compile in support for 32-bit ELF objects - you'll need that until you're fully transitioned. (You didn't uninstall the existing 32-bit kernel, did you?)
This system should happily run your existing 32-bit binaries. You can now install the 64-bit build of
dpkg; this will now make amd64 be your default architecture.
Then upgrade, a few packages at a time (unless you're
foolhardybrave enough to risk an all-at-once upgrade), and you'll have a fully 64-bit system.
It was a bit tougher when I made the switch, before multiarch - I had to take the big hit due to library dependencies, and it went badly wrong part-way through. You may want to move to a more recent Ubuntu (perhaps 14.04 LTS) while still running 32-bit everything, and then migrate. That's more likely to be successful than the opposite.