Is it possible to get cancer if you injure yourself from a needle used by a cancer patient ?

It would indeed be a unique situation if cancer were passed from patient to patient through a needle prick. Suppose we shouldn't say it could perhaps happen, as we are not supposed to say "NEVER". But being unaware of this happening in the past. Cancer is not an infectious disease.

Cancer is not transmitted through needles, however viral infections, such as hepatitis B or C can be transmitted through needles and they may predispose infected individuals to liver malignancies.

No. What you described would not cause cancer. You can reduce your risk of cancer by adopting healthy habits: Eat a mostly vegetarian diet, avoid alcohol, tobacco and street drugs, avoid excessive exposure to sun, maintain healthy weight, get HPV vaccine, exercise 30 minutes each day, drink enough water so your urine is mostly colorless. Wish you good health!