Don't try center drilling with it just sticking out with no support it will flop all over the place and your center hole will not be on center.
Putting it up the spindle so it is held close to the end in the chuck is okay if your chuck is perfectly true or if the entire length will be turned on the OD.
Sometimes the most precise approach is to do the center drilling with the end in a steady rest. This way the end will be running true because the material is being supported on the OD.
Even with the part supported in a steady you can get the hole off center if the center drill starts to wobble as it enters. For really picky work sometimes you may go to the extreme of boring the taper in the center hole with a really tiny tool from the compound slide after you have drilled.
An open mind is a virtue...so long as all the common sense has not leaked out.