I am looking to extend my apartment lease in NJ , and I want to know if I can include an addendum which allows me to vacate the apartment at one months' notice without incurring penalty. I once heard from some one that such an addendum can be added, if I pay some money upfront. Can someone validate this ? Does the lease office have the right to reject such a request ?

There is no law in NJ that allows or doesn't allow this type of addendum. It is entirely up to the landlord if they are willing to allow it. And if the landlord is willing they can charge whatever fee they want for it. Speak to your landlord to see if they are willing to do it.