Can I use old bits for a hand-powered drill in a power drill?
If they are HSS quality bits they would still be usable. Try by first drilling into wood. Then use a drop of oil and try on an aluminum block. 

Old bits are of great help with wood and plastics. Rarely would they break up and damage anything as long as you take care. Wear safety goggles. 

By the way...as they cost a bomb in India, I regrind broken bits and temper them again for general workshop use. With care the bits in the picture would last you a lifetime. I still use an ancient HSS 1/8 that has stood by me over the last 30 years! Hope this helps....