I prefer to drill on the lathe, but I usually use a metal drill bit. A brad point db is made to go across the grain; along the grain, it will love to follow it. Never have problems with metal db, only it heats up a bit more maybe.
If the shape of the piece to be drilled is too awkward, I mark where I want the hole to start and ends, I make a shallow hole freehand with the db at one end, I chuck the db in the headstock, the end of the piece with the little hole touching the db, the other in the tail stock, I hold the piece with pliers and the hole is exactly where I want it to be. The blank has to be a little longer than the tube. I find it useful too for some segmented blanks and antlers.