Does VirtualBox run 64-bit guests on a 32-bit host?

Your CPU does not provide VT-x harware virtualization needed for running a 64-bit guest OS in a 32-bit host. There is nothing we can do to overcome this.

In order to still be able to run a 64-bit application (as was said in a comment) let me suggest you run Ubuntu 64-bit either by a dual-boot, or as a live system from USB. A live system in persistence mode allows to save files in a reserved area.

Alternatively it is also possible to completely install a bootable Ubuntu on a USB drive/key:

• How do I install Ubuntu to a USB key? (without using Startup Disk Creator)

Support for 64-bit guests requires CPU virtualization support in the host, A.K.A. VT-x (or AMD-v). You need to verify that your CPU has this feature.

Navigate to BIOS and enable it if it’s supported by your CPU, try to google your CPU Specs.

Conditions for running 64-bit guests on 32-bit hosts:

VirtualBox supports 64-bit guest operating systems, even on 32-bit host operating systems, provided that the following conditions are met:

• You need a 64-bit processor with hardware virtualization support (see Section 10.3, “Hardware vs. software virtualization”).

• You must enable hardware virtualization for the particular VM for which you want 64-bit support; software virtualization is not supported for 64-bit VMs.

• If you want to use 64-bit guest support on a 32-bit host operating system, you must also select a 64-bit operating system for the particular VM. Since supporting 64 bits on 32-bit hosts incurs additional overhead, VirtualBox only enables this support upon explicit request.

• On 64-bit hosts (which typically come with hardware virtualization support), 64-bit guest operating systems are always supported regardless of settings, so you can simply install a 64-bit operating system in the guest.

• Warning

  On any host, you should enable the I/O APIC for virtual machines that you intend to use in 64-bit mode. This is especially true for 64-bit Windows VMs. See Section 3.3.2, “"Advanced" tab”. In addition, for 64-bit Windows guests, you should make sure that the VM uses the Intel networking device, since there is no 64-bit driver support for the AMD PCNet card; see Section 6.1, “Virtual networking hardware”.