Khaosan in Asakusa is 5000 yen for two.
The room is small but it is reasonable for the price.
Also, hostels are great way to go.
I know one place in Yoyogi which is right next to Shinjuku, for only 3000yen.
Japan Youth Hostels, Inc.
Only downside is they have a sigle bedrooms only so you won't be able to share a room with your partner.