I’ve seen some articles recently that talk about this exact question (because it’s all related to the movie). I’m at work so can’t look any up now, but if you keep on accelerating at 1g that time dilation kicks in. You can reach Andromeda within something like 15 years and the edge of our observable universe in about 33 years. It’s wild to think that we could get that far in a human lifetime.
Of course, the longer you keep it up, the more magical a constant-acceleration drive needs to be. A constant 1 g from here to tau Ceti is just maybe plausible, with an antimatter-based fuel. To the Andromeda galaxy, though, even with pure antimatter fuel, the fuel ratio (amount of fuel needed to total payload) gets ludicrous, and for the edge of the observable Universe (as observed from Earth right now)? Fuhgettaboutit.
If you can maintain 1g acceleration for a year and stop accelerating, you’d buzz past Tau Ceti at roughly 0.77c, give or take. It would take roughly 16 years to get there, as measured from earth. Ship time around 10 and a half years, time dilation still being significant at that velocity.
Longer if you decide to decelerate and orbit, of course.
Well, assuming you can maintain that sort of acceleration indefinitely, sure, as least in ship time. But that’s in the same realm as unobtanium and spherical cows as far as we know.
The math is interesting regardless of the engineering 
One interesting option I’ve seen is that the starship drive opens a tiny wormhole into a newly expanding universe, so you can ‘borrow’ power from a Big Bang somewhere in a different cosmos.
This would give you enough power to accelerate at one g for an indefinite period; but itt doesn’t solve the problem of shielding. At any velocity faster than 0.85c the energy imparted by a gram of dust would be equivalent to a gram of antimatter hitting the ship. Interstellar protons become deadly radiation, etcetera.
Also all that energy in your rocket motor would heat the ship - even at 99% efficiency there would be waste heat, which would melt the rocket quite quickly at these energies. You’d need massive radiators - which would also need to be shielded from interstellar dust and gas.
I don’t know how PHM deals with these problems.
I once calculated travel at 1G for an sf story. IIRC, I found that the time in earth years was always 1 year off the number of light years traveled. It’s been decades so please confirm if this memory is correct.
Certainly not that simple, or it’d take 1 year to travel 0 distance. But it’d be a reasonable approximation for long distances.
Yes, obviously for star travel.