The property you found is exactly as you described, elm ridge park is technically in hopewell and a lovely community of larger homes that are not cookie cutter in design, elm ridge park is about 10 minute drive to down town Princeton.
Princeton itself has very few rental apartment buildings. those that exist typically have a 2 bedroom apt as the smallest unit and for more money than is in your budget. For your price you may be more successful looking through the local paper resources and websites for privately listed residence, rooms etc including - Craigslist, or rent.com ( as a realtor I dislike recommending this approach but often times people who a have 'wants' like yours are more successful this way)
Optionally, you may find a 1 bedroom apartment in plainsboro or Lawrenceville for that amount of money. These towns are also within 10-15 minute drive to downtown Princeton.
Good luck !
I feel your pain. I am not sure what you mean by the Princeton Area. In Princeton itself and most of the immediate towns surrounding Princeton you choices will be very limited with your budget. Those places that you do find will probably not meet your expectations.
So what to do? Expand your search radius and agree to travel a bit more to get to work or except what you can get closer as the norm and make a decision on the best the pack.
The rental market in the Princeton area has very little difficulty finding tenants. One must align themselves with a Realtor who is willing to do the grunt work to give you advanced notice of an opportunity thereby you will be one of the 1st to see and make a decision.
Finally, I would suggest you look at some of the manged sites that are 9-10 miles away from Princeton proper as I think they will be close to what you want to spend and the upkeep is stellar. What you will have to give up is some size.