Can I have backup power for an Arduino device?

you could set up a relay to work in reverse with the mains power,

have your backup battery connected to the gate contacts of the relay, and then use the mains power running your project to open the relay and keep it open as long as there is power. if the mains turns off, then the relay closes and the battery is connected.

you might want to put something in line with the power to the arduino to make sure you don't get any spikes when things switch.

hope this helps as i love kittens

I suggest this "Battery boost" circuit from AdaFruit. It's designed to do exactly what you describe. An Arduino needs a smooth, steady 5 V supply with no "blips". The AdaFruit unit does just that, and charges the battery while on Mains supply.