Let's imagine an infinite Universe with just a single particle in it. If there is nothing else in that Universe, there is no reference frame relative to which the particle can be considered in motion or stationary. In this case the motion cannot be defined. Would time still exist as a more fundamental propery of that Universe or it will be also undefined? How would time be defined in this Universe?
Is there some physical phenomenon in this toy-universe that requires a time dimension in the theories describing it? If no, then time is not physical in this universe. You can of course introduce time in any timeless theories, but it would not add any explanatory power...
To a certain degree, this is no longer a philosophical question, but one of physics.
From a very narrow modern physics interpretation every particle has a wavelength. The particle 'vibrates' in the sense of the wave model of matter because matter is a form of energy, subject to Heisenberg's principle. It must move, or we would know its exact extension too well. So time could be defined in terms of that vibration.
From a broader application of the same theory, this universe would not exist for very long. Virtual particles will arise from any vacuum in pairs of material particles and antiparticles. So your problem of not having other landmarks would solve itself.
The nature of time in a single-particle universe
Julian Barbour writes in The End of Time:
The first step ... is an inertial clock. The German mathematician Carl Neumann took this first step to a proper theory of time in 1870. He asked how could one make sense of Newtons claim, expressed in the first law of inertia, that a body free of all disturbances would continuer at rest or in straight uniform motion for ever. He concluded for a single body by itself such a statement could have no meaning. In particular, even if it could be established that the body was moving on a straight line, uniformity without some comparison was meaningless. It would be then neccessary to consider at least two bodies.
However, two millenia before Carl Neumann a very similar question was asked by Aristotle in his exploration of the meaning of what constitutes the void in his Physics and why it cannot exist; he argues the existence of the void implies that the particle must move in all directions at once as no one direction is singled out (this is an argument from symmetry) and as this is not possible the void is not possible.
Now, it sounds strange for a body to move in all directions when we generally consider that particles move in a definite direction (as we see from observation); however, this is classical mechanics; in quantum mechanics, we can describe the motion of a single particle in roughly this way (the calculation of the evolution of the wave-function of a single electron is a standard calculation in undergraduate physics); this of course, isn't to say that Aristotle invented quantum mechanics (after all he disputed this form of motion) but merely that a conceptual investigation of what constitutes matter, motion, place & void was touched upon and investigated vigorously in Antiquity.
He didn't, as far as I recall, touch on the problem of time in this setting; at least not directly.
Indirectly, we can say more; Aristotle, pointed out that there is a close connection between time and change, of which the primary sense is motion; Julian Barbour in his book suggests that time is a derived or emergent concept, and that change itself is more fundamental; as the wave packet certainly changes we can speculatively say that time exists even in this situation.
Now, the mainstream Copenhagen interpretation allows only classical measuring instruments to collapse the wave; since there is none such around, there can be no measurement to collapse the wave and actualise it; hence we might say that in this situation that time exists potentially, but not actually.
But more can be said under the Relational interpretation where collapse can be triggered not just by a classical measuring device but by any system, including another particle.
For, whereas single particle systems are possible in QM; when we add relativity we get the theory known as QFT; and here, single particle systems are not possible, they are always many particle systems - so the single particle is surrounded by a swarm of virtual particles; here, collapse can occur, so we can speculate that time emerges actually here.
An universe of one particle assumes at least two entities: the particle and the non-particle stuff. In addition, the configuration (a particle), assumes a differential n-dimensional space. The lack of other particles does not imply that the particle would not move. You will ask "movement, in relation to what point of view?"
The problem here is the definition of movement: a change in a temporary-spatial configuration is a rational assessment of some change (that we still do not understand): without a human observer, not only movement is not possible; time and space don't exist without a human subjective assessment (see Kant, regarding time and space). Movement is a human, rational subjective approach about changes of an object.
In addition, particles are a rational understanding of some natural context. Particles don't exist in nature. The real universe is about empty space filled by quantum fields. Particles are some manifestation of quantum fields which we perceive as our reality. All mass is just interaction (Feynman).
The idea of a unique particle in empty space is pretty naive. Has a particle some volume? If so, it can easily be named a planet... How would an isolated entity exist without interacting with others? Only what can be perceived exists (Berkeley). How can a particle exist without a systemic configuration as a group of interrelated parts (this is the same question as the previous ones, just reformulated from the systems theory perspective)? If so, this is our known nature, with your body on the center of the universe: while you don't leave the particle (the whole big bang debris) you can measure any movement.