I just found 53 rentals in East Brunswick (08816) on realtor.com rental as you can see ...
http://www.realtor.com/search/search...p=08816&typ=40 Do not hesitate to call or email the agents who may know more detail and may be other units available in the near future ...
I also found some in craiglist.org central jersey east brunswick rental
http://cnj.craigslist.org/search/apa...100&bedrooms=1
In particular, this one seemed to match 110% of your crieria, only $1,000 !! Yeh! http://cnj.craigslist.org/apa/518448951.html (broken link)