You need to know the total current requirement to figure that out.
My Arduino draws about 25 mA while running just about any code that doesn't light any LED's.
An Energizer 9V alkaline battery is rated at 600 mAh or "mili-amp hours" with a 25 miliamp load - so 600 / 25 gets you about 24 hours. One problem though is that they rate their battery at 600 mAh down to 4.8 volts. The Arduino regulator will stop putting 5 volts out once the battery drops below 7 volts so you should probably cut that 24 hour calculation in half.
when needing to run from batteries, I run my Arduino from a 6 AA cell battery holder which has a typical 9V battery clip type connector on the end, I fill it with 6 NiMH batteries. Arduino circuits are a little power hungry for a little 9V alkaline.http://search.digikey.com/scripts/DkSearch/dksus.dll?Detail?name=BH26AASF-ND