I actually found Japan very cheap compared to Europe and the USA, but it depends what you are into I guess.

Personally, I would stay away from the popular tourist traps, but there are many places you could go out and spend very little.

Osaka Castle area is a perfect place to walk around the gardens, have a picnic lunch and enjoy the company you are with. While in the area you could always spend a day at Spa World if the weather is not best. It does get very busy on weekends with the natives, they do have a hotel in the complex.

Tokyo I found you really need to get away from city area, Harajuku/Yoyogi park especially on a Sunday is well worth a day and always changes, if you get bored you could always move onto Shibuya for some window shopping/lunch, also known for Love Hotel Hill

Asakusa, has a great market atmosphere and Kannon Temple with many cheap places to eat, cheap shopping, but a little too western for my liking.

At the end of the day you could go all out and do the expensive things, or enjoy each others company and walk through the many gardens getting to know each other. Dates don't have to be always night time, maybe an Onsen Inn weekend away would suit you better.

Osaka/Kobe/Kyoto I found romantic, Tokyo I found was just a messed up city trying to be like any other Western city, but hidden area's could be found.