MC if your lathe has a hollow tail stock try drilling through it and making a pilot hole just as a guide. Another way is before turning your blank round clamp 2 bits of wood on adjacent sides and use them as a guide for your hand drill. Or turn and hollow a piece of wood that can fit over the end of your blank but is short enough to go between centres and drill and then use that as a guide. There are loads of ways to achieve the results you want, work out a way to drill the blank before turning it round, so I would hold the drill bit in place and then think what I need to do to keep it running true.
Hope this helps.