Can I get infection if med student made lots of small puncture wounds on back of wrist with same IV needle?

Potentially yes. A needle can become contaminated after the first use. Any subsequent use whether accidental or intentional can lead to many types of infections. Usually only skin related contamination occurs, however, transmission of viruses, i.e., hiv, hepc, etc, is possible. The larger the size of the needle, the greater the risk. Needles should never be shared.
Not if skin cleaned. I am guessing you mean that each of the wounds was created with the same needle, not that the needle was used first on someone else. That would be the only way to give you "germs" from another patient and the only way that the diseases mentioned in the other answer could be transmitted to you. Sticking the same patient several times with the same needle can only bring in germs from their own skin.
Possible. Your skin and mucous membranes are the major immune defense mechanism your body possesses. When you disrupt that barrier you may introduce organisms into the body that have the potential to produce infection. I suspect the student was attempting to start an IV site or to draw blood from a small vein unsuccessfully. It is not uncommon to do several sticks with the same needle, but prefer change.