If you don't mind turning on arduino every time you will likely push buttons, adding a power switch will save your battery. I have two rechargeable 9V batteries (very weak batteries, don't buy them!). If I keep one project on battery, it runs out in less than two days. For the same project I have with a power switch, I occasionally turn it on and it still works after about 2 years.
The clock on the shield MichaelMeissner mentioned has a backup battery. That battery will last years.