Don't know how you got that figure. First of all 200mA per day is meaningless, I think you want mAH per day.
If the pump draws 350mA and is on 8 minutes per day, this is 8 / 60 * 350 = 47mAH per day. Rechargeable AA NiMH batteries (about 2000mAH) would run it for about 42 days, so you...