Do the best that you can to align the tailstock, but don't be disappointed if you don't get a perfectly straight cut on a long piece. For the most part, a thou taper is typically good, and may or may not hold steady at that value. There will always be residual spring of the workpiece, plus, if the part is flimsy, and you need to use a steady rest, you can easily correct or force a bend into the part, which will give you a taper. Each part is more of a science experiment than many realize, and a well experienced machinist has learned all the tweaks as if by rote. CNC is only wonderful in the way that the 'experiment' is carried out in a very repeatable fashion at near optimum cutting speeds with tools in near mint condition (which stay that way because they are run near optimum speed with near zero dwell), so the tip suffers a very low rate of damage.
I seldom have to turn very long straight cylinders (thankfully) and my lathes are in good condition. But different material hardnesses affect the springback, so what is a good setting for a long piece of aluminum tube won't be the same as for turning a large bar of 4140. You'll have to learn to master the skim cut with a very sharp tool, or incremental cut adjustments along the way as you try to compensate for the spring.
In the long run, you will be filing and polishing the part to some degree, so equip yourself to do so. Air powered belt sanders can be had fairly cheap, and those are good for sizing and removing toolmarks. But, handling the sander is yet ANOTHER art to learn, depending on the grit, how fresh the belt is, how fast the part is revolving, etc, and how fast your lead is across the part, near the ends and near the shoulders. I do like finishing with about a 180 grit sanding belt, which when fresh, takes off a thou or two rather quickly, but then after that, it kind of settles into a more stable finisher. It may leave what you consider a suitable surface as is, or you can further shine it up with finer grits carefully held in a rig or by hand. But once you've sanded a uniform finish onto a fresh turned part, then it becomes a lot of work to remove all the material for a further sizable reduction. A full thou then seems like a mountain to remove by hand polishing.
Wear a dusk mask while sanding, or ventilate to the outdoors. We all know how important having a good set of lungs has become.