Hi,
All I can say from my 2 visits in Ethiopia is that Addis is a pretty save city, even during the night. I would even consider the Piazza area save at night.
For a taxi from the airport to Piazza 30-40 Birr is a good price as taxis are more expensive at night time. The best place for bars and restaurants is definitely Bole road - there's plenty and it's not far from the airport. Nearly the whole stretch of Bole Rd. from the airport to Meskel Square is full of bars, clubs and restaurants.
One of my favourite is Habesha restaurant in Bole Rd. They serve excellent traditional food and you'll enjoy the traditional entertainment. Have a look at: http://www.habesharestaurant-addis.com/
They're open til 2 a.m. and I think even on Sundays.
If you're after party and dancing try the Platinum Club: www.platinumaddis.com
Have fun !