I stayed in Japan(Tokyo, Hakone, Nagano, Kyoto, Osaka, Hokaido, etc) for 5 yrs back in late 80's - early 90's and went back to Kyoto last 2week ago for Autumn leaf(disappointed) but I have fun and fond memories of yester years.
Sakura season is from mid April to May but there is early/late season every year depending on weather and location. Safe bet is end of April, I guess. Best park to view Sakura will be Ueno park in Tokyo. You can take Yamanote Line train ( this train run in circle around Tokyo city) and stop at Ueno station. It's just a few minute walk from the exit. Warning: Do not get lost in this huge and amazing station. Goggle Tokyo train map. It's the green line in circle/oblong shape at the center.
If you and your wife are adventurous, I suggest you free and easy tour. There is more to discover in and around Tokyo and within your budget. For staying, you can book into any Ryokan(guest house) which is cheaper and experience the life of Japanese culture. Food is everywhere in Tokyo. Just go in and point at menu of what you want to eat (assuming you can't speak Japanese). Don't worry, they won't slash throat, knowing you are foreigner.
If you need any contact in Japan (Malaysian who are working in one of Japanese IT company for 6years), I can give you my friend contact. I'm not sure of any charges but you can deal direct with him through email.
Pssssttttt: I can be Japan tourist ambassador/guide. Anyone, want to buy my stay in Japan. Lol
