Can I install a 64-bit version of Ubuntu on my 32-bit machine?

By "32-bit machine" I understand 32-bit hardware (processor) which is not your case because lscpu reports that your CPU can work in 64-bit mode.

CPU op-mode(s): 32-bit, 64-bit

You have a 32-bit software (operating system) running at the moment on your PC, that's why:

Architecture: i686

But you can install a 64-bit operating system (you have both a compatible CPU and enough RAM for it).

You can install a 32 bit system on 64 bit hardware.

The converse is not true.

You cannot install a 64 bit system on 32 bit hardware.

From the output of lscpu you gave:

CPU op-mode(s): 32-bit, 64-bit

It looks like your hardware is in fact 64 bit. You can install a 64 bit system.