I tried to post details of a great apartment for short lets on the accommodation section, but Tripadvisor wouldn't accept it. Apparently they do not list individual rentals.
Anyway, if you are looking for a furnished apartment to stay for a few days in Neuchatel, have a look at the Loft (www.loftus.ch). We stayed there for 8 days at the beginning of August. It's a super apartment, very room, very comfortable, and well-placed for getting into town.
Most places we have stayed in look great on the website, but when you arrive they don't look quite so good. Not so with the Loft. the reality is absolutely equal to the description and pictures on the website. It's definitely the best apartment / cottage we have rented.