[chachava]

is your budget for a week, or a month? If for a month, you might have to look outside of Tokyo and commute - I pay about just under 350000 yen a month for my appartment in Yokohama... the further out you are from a city centre, the cheaper it will be however

My advice would be to maybe try looking in Kawasaki or even Tsurumi as they are a little further from Shinagawa, but you only need a single train (the JR negishi line will go straight to Shinagawa) to get there which would be quite easy for you

No idea about furnished appartments... mine had nothing when I moved in apart from floors, doors, shower and a toilet lol