To my best understanding, in a "big rip" scenario the universe expands faster and faster, until it expands between particles faster than the particles can hold one another.
Does that mean that at some point atoms reach temperature of 0 Kelvin, because the universe expands so fast, there is no way to actually "move" anywhere? How does that work with quantum uncertainty?
Velocities of particles in an unbound motion in expanding universe would decrease in time. The intuitive understanding for this effect is easy to gain if we consider de Broglie wave of an unbound particle: as the universe expands so is the de Broglie wave of an unbound particle, which means that the de Broglie wavelength of a particle is proportional to the scale factor of the universe: $\lambda (t ) \sim a(t) $. For nonrelativistic particles this means that $v(t)=\frac{p}{m}=\frac{h}{m\lambda (t)}\sim a(t)^{-1}$. Such reasoning also has an uncertainty principle "built in": as the motion slows with the universe's expansion, uncertainty of the position increases correspondingly, since now there is more expanded space for a particle to be in. Kinetic energy of an unbound motion would decrease as an inverse square of a scale factor $E_\text{kin}=mv^2/2 \sim a(t)^{-2}$.
Motion of particles within bound systems (from electrons in atoms to galaxies in galactic clusters) would remain mostly unchanged by the universe expansion as long as the Hubble parameter remains around the present value. That is why for example, current expansion of the universe has almost no effect on orbital motion in the solar system, see this answer by Ben Crowell, as well as multiple other questions on this site.
However, in the Big Rip scenario eventually all bound systems would be torn apart by the expansion of the universe, first the largest and more loosely bound and then smaller and smaller. Eventually all matter would be torn into unbounded elementary particles, which would then be further and further slowed down by continued expansion.
Also note, that usually, when talking about temperatures we (sometimes implicitly) assume a certain thermodynamic equilibrium even if it's approximate and only achieved locally in a small volume. In case of the Big Rip once a particle becomes unbound it would likely never again interact with anything else besides gravitational field, so while the energy of each particle would be rapidly approaching zero the overall energy distribution would not be thermal, so speaking of temperature would be imprecise.
In an expanding universe co-moving peculiar velocities (i.e. velocities measured relative to a coordinate system where objects that merely drift with the expansion of the universe; this coordinate system will stretch with the expansion) will decrease as $v_0/a(t)$ where $a(t)$ is the scale factor. So in a big rip the co-moving velocities do indeed disappear - but this is because the distance measures scale up.
The proper peculiar velocity (i.e. the velocity measured relative to a coordinate system that doesn't stretch with expansion) will remain constant. A particle with a certain momentum will not lose its momentum, and kinetic energies stay the same - so there is no change in kinetic temperature.
Finally, quantum uncertainty isn't a problem even in the usual zero Kelvin case: it just says that free particles will get a very uncertain position (see the answer here).
