Assuming we have the launch velocity of the arrow (which I will take to be 82m/s (184mph)) and the force of gravity (9.8m/s) we can use the equations of motion to calculate the maximum height of the arrow.
The equations we will use are:
v is the final velocity,
a is the acceleration, v0 is the starting velocity,
r is the distance from the starting position and
t is time.
The actual calculation:
We want to find the maximum height the arrow will get to, at that point the velocity will be 0, so
v=0. We are ignoring air resistance so
a is -9.8m/s. With that we can find the time at which the top of the arc is reached:
Now we can plug that into our second equation to find the height at which that occures
So could an arrow get there?
Absolutely, ignoring air resistance the arrow could get to the top of the wall and keep going for a further 130 meters. This would clearly be reduced by air resistance (and not shooting straight up) but I'd be confident it would get there.
Ok, so it gets there, but how deadly would it be?
As the arrow climbs it slows, and the slower it goes the less dangerous it becomes. We can again use the equations of motion to find how fast the arrow would be travelling at this point
v=50m/s (112 mph)
In other words still pretty damn fast.
However the energy that the arrow delivers is proportional to the velocity squared, as we know by Kinetic energy=0.5*mv2. For this we will need a mass, I've used 300 grams but it turns out not mattering since I'm just creating a ratio:
Energy at launch=0.5*0.3*822=1000 joules
Energy at the top of the wall=0.5*0.3*502=375 joules
This is starting to look a lot less impressive, for every joule of energy the archer puts in at the bottom of the wall a little over a third impacts the armour of a soldier standing at the top of the wall.
Ok, the wildlings are in trouble, how deadly would the arrows from above be?
For this we just reverse the calculations above. Doing this we get the following arrow velocity when the arrow reaches the ground
v=104m/s (230 mph)
Arrow energy: 1622 joules
(Incidentally terminal velocity for an arrow appears to be around 110m/s so an arrow could go this fast, although as always air resistance will reduce the actual velocity)
The wildling arrows could certainly get to the top of the wall but they would have only 35% of the killing power of an arrow when launched. Furthermore the Night Watch's arrows would be around 60% more deadly than usual. While both parties would have difficulty aiming over such a distance the Nights Watch would find it far easier to penetrate armour. Additionally the wildlings would need to shoot at their greatest strength whereas the Nights Watch would need to exert less effort; this could impact accuracy.
Accuracy is ultimately going to be up to the skill of the archer, but it is worth noting that the slower the arrow is travelling the more its trajectory is affected by the wind.