Most of the nice apartments will require a payment of key money, which is basically equivalent to 1-3 months rent. Location/age of the building/prestige of neighborhood are all determinants as to how much key money you will have to pay.
If you can read Japanese, try looking up fudosan on Yahoo!Japan for starters. Also, Chintai magazine (sold in Japan and maybe available at your local Japanese supermarket in the States) has lots of cheaper listings.