What will that do to the speed of light as measured on Earth?
And will this affect, say, the correct operation of computers?
The dipole anisotropy of the Cosmic Microwave Background will continue to be a good tool to see the constancy of the speed of light for all observers into the practically infinite future, assuming the infinite future looks a lot like de Sitter space (which is essentially what the standard cosmology predicts).
http://www.astro.ucla.edu/~wright/CMB-dipole-history.html
The far future of the universe will still have generally covariant physics, as does its far past, sharpening up your second sentence.
As far as the measurability of the speed of light, I don't think anything changes in principle. It should remain measurable and constant. However, it gets a lot more complicated in practice, since it's pretty unclear who or what would actually be doing the measuring and how they might go about it.
We also have good views of the Lyman-alpha forest through galaxy clusters at different redshifts, and the distribution of the spectral lines are another line of evidence that members of galaxy clusters closer to us have smaller radial components in their peculiar velocities than those at higher redshift/smaller angle/lower brightness.
To get to a big rip you have to impose a further time derivative and figure out how to keep it small now and make it big later. AFAIK nobody who has tried this has avoided significant complexity to match the observational data we have today, except by pushing the Big Rip so enormously far into the future that we don't have enough data to constrain that future. The wikipedia article refers to WMAP9, which ended 2013 (~ 9 years after WMAP turned on in 2003). There is more recent data from Planck and other observatories, and they make it harder and harder to justify a Big Rip for the reasons half-admitted in the "Observed universe" section of that wikipedia article.
Moreover, MESSENGER and other spacecraft sent to parts of our solar system have given us a lot of data about the metric of our solar system, and it is non-expanding; orbiting clocks support this more locally. Likewise, there are plenty of sensitive terrestrial experiments that highly constrain any expansion term in plausible Standard Model + gravity Lagrangians.
It can't be ruled out but it has been effectively pushed back so far that it's the least of our worries about "what happens hundreds of trillions of years in the future" : if one accepts there there is a Big Rip field, there's no compelling reason why it should have significant effects on matter in the local group that soon, and several arguments about why it shouldn't.
You'll find a bit more substance (and higher difficulty) at https://en.wikipedia.org/wiki/Quintessence_(physics)#Tracker...
Hurry up Elon, we're running out of time!