You can certainly imagine such a Universe with one lonely structureless and cold particle, interacting with nothing and living in some eternal and unchanging spacetime. Of course such a Universe it is nothing like our own. My answer would be that since nothing ever changes in this Universe, there is not really an operational notion of time.
We can imagine this is a massive quantum particle living in Minkowski space; then there is at least a frequency associated with the Compton wavelength of the particle (in its rest frame), which would put a physical clock in the Universe. But there would be no arrow of time
Such a Universe is so weird that it borders on not being mainstream physics. But I can see its value as a thought experiment as in your question to clarify that, indeed, to measure time there need to be something changing, and to have an arrow of time there needs to be irreversibility -- in our Universe, this emerges through the second law because entropy was lower in the early Universe, and (on average) increases toward the future.