I use a chopstick or the spine of my knife blade and pick at the root ball to release the peat moss I grow stuff in and loosen the roots. Usually works great, but sometimes I have to cut. I don't generally give them enough time in a pot to get badly rootbound.
If I have to cut some roots, I usually cut some top growth off. I cut a bit more top growth than whatever roots I cut because trees seem to do better when unbalanced toward the roots rather than toward top growth.
If you have to cut roots on Thuja, cut the lower branches rather than the top of the plant. Thuja can send low branches from the trunk out under the lowest growing branches and retain its natural growth habit. I only have experience with wild Thuja and Smaragd (Emerald?), not with Green Giant, but I would guess they are similar.