OK, so we are talking about a 3/8" hole right around 4"+ long.
As Ken suggested, first turn the blank round (between centers) and drill on the lathe, preferrably using an appropriately sized collet (9/16" or 3/4").
If you are really careful, you can do it without a collet by gripping the blank end in your headstock chuck. . Hold the drill bit (which will need to be longer that 5" or so, and I think the standard jobber drill will do it) in a Jacobs chuck in your tailstock, and with the drill tip carefully centered on the free end of the blank, start drilling.
Keep your left hand loosely gripped on the rotating blank and drill away.