How long would it take to get to the Hyperion proto-supercluster? Right now, we set a ship carrying no humans going to the Hyperion supercluster (redshift is $z=2.45$, so around $11$ billion light years from Earth) at a velocity of $c/2$ ($c$ being the speed of light), accelerating as quickly as possible and decelerating as quickly as possible right before arrival. 
How long would it take to get there, not in terms of the relativity of the ship, but the age of the universe, assuming "right now" the universe is $\sim 13.8$ billion years old. 
For the most distant objects in our observable universe, I assume expansion of the universe and movement of astronomical objects will have a lot of effect on the time it will take for the ship to arrive, but I'm not sure how to calculate this. I am guessing I need to calculate the comoving distance, but I'm not sure how.
 A: The answer is never. The reason for this is the accelerating expansion of the universe.
The Hyperion supercluster is 19.171 Gly away in terms of co-moving distance according to CosmoCalc. In a non-expanding universe travelling there at half lightspeed would take just 38.3 billion years, but while travelling that time the proper distance will have increased by roughly a factor of 15! Going at lightspeed means the distance still increases by a factor of 4. This is because the universe is expanding at an accelerating rate, with a scale factor going roughly as $a(t)\sim e^{t/H_0}$ where $H_0$ is the Hubble time (this is an approximation that gets better in the future, assuming the $\Lambda$CDM cosmology is right) . 
We cannot reach beyond a certain range from Earth today even when travelling at lightspeed (the cosmological event horizon); this distance is currently about 13 billion lightyears. The (proper) distance to it at time $t$ is $$d_{EH}(t)=c a(t) \int_t^\infty \frac{du}{a(u)}$$ (the co-moving distance drops the $a(t)$ term in front). This is $ \approx cH_0$ today. 
So when will you arrive at an object that you can reach? If you boost your spacecraft up to some velocity $v_0$ and then coast, your peculiar velocity (the velocity you measure relative to galaxies you are passing by) will decrease as $v(t)=v_0 e^{-t/H_0}$, which when integrated becomes $\chi(t)=v_0 H_0 (1-e^{-t/H_0})$. This is in co-moving distance, the distance measured proportional to the number of galaxies you have passed (most matter in the universe stays at the same co-moving coordinates). The corresponding proper distance (the distance you would measure at a given instant by putting yardsticks from your starting location to the endpoint) would be $x(t)=\chi(t)a(t)$ and would be growing exponentially. Turning it around, we get $$t_{reach}=-H_0 \ln\left(1-\frac{\chi_{target}}{v_0H_0}\right) = H_0 \ln\left(1+\frac{x_{reach}}{v_0H_0}\right).$$ The left case is for a given co-moving distance which is usually what you want: as you approach the horizon the time to get there diverges. The right case is for proper distance: just getting far from Earth is easy, just move at any speed in any direction and wait until expansion has increased the distance a lot. 
It is pretty cool that there are places we can observe that we can never reach (barring FTL) or even influence. It is also mildly annoying. 
