Run 64-bit app on 32-bit Ubuntu system

You can't do that directly on Linux on x86_64 with a 32bit kernel. You need a 64bit kernel to be able to run 64bit code. (Note: kernel. You can have 32bit userland running on a 64bit kernel, and install support for 64bit binaries. How easy this is depends on the distribution. Example here for Debian.)

Some virtualization software is able to do that (with restrictions on the type of CPU you're using), and emulators can do that too. But to run it natively you'll need a 64bit kernel.

Since most 64bit distributions out there have 32bit support (either by default or installable), that's what you should be using if you need to run both 64bit and 32bit binaries on the same OS.

Extending @Mat's answer:

Yes, some virtualization software can do it. The main obstacle to handle is that in a 32-bit environment, the 64-bit features of the CPU are not available, and also the features of a 64-bit kernel isn't available. It is even so if your CPU has 64-bit support (since around 2003, all of them have).

Your question implicitly states that you want to run the 64-bit app in your own system environment. If it is really your intention, this excludes all virtual machine-based solutions. The only remaining which is known for me is the kernel-emulation of the qemu. So:

qemu-x86_64 ./your_64bit_app

It runs your app on your current system, with your current kernel, with your current environment variables. Of course it is not very fast, it has to emulate your cpu.

It can emulate only linux x86-64 apps.