Does the expansion of the universe make time pass faster? Afaik. mass slows down time, while expansion means that the universe will be less dense in mass. Does that mean that time will go faster everywhere?
 A: Your question assumes that there would be a absolute time or clock that we could compare to.
There is no such absolute time. 
It is a misconception that mass causes the effects of gravity and time dilation. In reality it is stress-energy.
Even if you are inside the event horizon of a black hole (where stress energy is very strong), you will see your own clock to tick normally for you. You have to compare your clock to a clock outside the black hole to see that your clock ticks slower then the clocks outside.
Now in your question you are saying that with the expansion of space, the universe will be less dense in mass. This would only be true on average.
Space is only expanding in the intergalactic voids. Space is not expanding inside galaxies, because inside galaxies, gravity dominates.
So if you look at the mass density inside the galaxies, you will see it is not becoming less dense.
It is just that galaxies are becoming more isolated.
Now in your case, if you would compare your clock here in the Milky Way to the clock at another galaxy, you would see that the clocks tick differently, but the difference (time dilation) would stay the same despite the expansion of space.
