For a reasonably priced room and convenient location, try the Shinjuku Park Hotel, just a 3-minute walk from the JR New South Exit of Shinjuku Station. Rooms are small by Western standards, but they are clean, stylish and have the basic amenities, including broadband internet in every room.
It is primarily marketed towards the business or single traveler, with 165 single or semi-double rooms, as opposed to the 7 double or 30 twin rooms available. Facilities are limited, but if you are only looking for a place to sleep, this hotel is recommended as a good value for the price.
For more information: http://www.shinjukuparkhotel.co.jp/english/index.html