I'm working out of Abuja at the moment, driving the airport road sometimes four times in a day.
There is good security along the airport road during the day with prominent army and police patrols. There is less security at night - don't travel at night unless you absolutely have to. Don't draw attention to yourself - eg remove your slides, keep windows closed. Biggest danger on the airport road is completely crazy driving (e.g. cars on the wrong side of the motorway are common) and pedestrians running across the road. RTAs are common (especially at night) and you won't get any help if you are involved in one.
The hotels are safe enough, and you can move around in Abuja during the daytime ok.