I don't know about apartments, but if you're flying solo, "First Cabin" might interest you. It's a budget type hotel (shared bathrooms), but really nice looking digs.
Try searching in Airbnb.com or Roomorama.com. They're booking sites that include apartment rentals. Whats nice about these sites is that they have certified photographers who actually go these places so what you see is what you get so how it really looks like in the room and how the neighborhood looks like outside is pretty accurate
If only Jonathan!...My eldest Son and I did two trips originally, then 2 years ago I took his 16 year old as well then earlier this I took my wife as well! She absolutely loved it! So no more solo trips to Japan for me!