Recently, fewer and fewer places are charging 礼金. There are some major rental agencies that won`t list apartments where the landlord charges it. It`s taking time, but I do think it is on the way out. The standard agency fee is about a month`s rent - at least around here and in my experience in finding apartments for people. It`s still about 3 or 4 months in advance though, but the law DOES support you getting all of your deposit back - regular wear and tear does not count against it, so if your landlord won`t give the part you paid as a deposit back... You can contact a lawyer. Usually just mentioning action will make the landlord have a change of heart and "remember" that the deposit was only for abnormal damage.
In my experience, landlords in the US are EXACTLY the same when it comes to this. They tend to conveniently forget that the deposit wasn`t to cover normal wear and tear until you remind them.
Anyway, to get back to the original question.
These are just a few links from Japanese real estate agency sites - by the time you arrive most likely none of these will be available. They should give you a basic idea of what is available in your budget and space desire range.
東京都品川区東大井3丁目京急本線 立会川駅 徒歩1分の賃貸物件情報｜お部屋探しならHOME'S賃貸
東京都品川区大井6丁目ＪＲ京浜東北・根岸線 大森駅 徒歩8分の賃貸物件情報｜お部屋探しならHOME'S賃貸
東京都品川区中延1丁目都営浅草線 戸越駅 徒歩7分の賃貸物件情報｜お部屋探しならHOME'S賃貸
東京都品川区小山4丁目東急目黒線 武蔵小山駅 徒歩2分の賃貸物件情報｜お部屋探しならHOME'S賃貸
東京都品川区西大井3丁目16-26ＪＲ京浜東北・根岸線 大森駅 徒歩12分の賃貸物件情報｜お部屋探しならHOME'S賃貸
東京都大田区西蒲田7丁目ＪＲ京浜東北・根岸線 蒲田駅 徒歩9分の賃貸物件情報｜お部屋探しならHOME'S賃貸
My personal favorite that I spotted was this one;
東京都品川区旗の台6丁目4東急目黒線 洗足駅 徒歩4分の賃貸物件情報｜お部屋探しならHOME'S賃貸
The bedroom is upstairs, and it`s quite spacious.
Those are just a few examples out of the 20000+ listed fitting your requirements - all the links I have listed are to apartments within 20 minutes commuting time of Oimachi, all 35m2+, and all relatively new buildings. Oh, and of course all below 130,000/month including all monthly fees.
There are tons of apartments out there.