Looking on Wikitravel and other trekking sites, it looks like much of the mountains are at least a couple of days away.
However, Entoto Mountain is probably your best bet. From the 'Do' section of Wikitravel:
walk from St. Mary's church, the first church of Addis and St Urael
church and see the city itself from the top of the mountain. If
planning to hiring a car - Never pay more than 300 birr for a taxi
unless you hire them for the whole day and are making other stops.
Walking is a great way to make it, but will take a good half day.
So there's a half day walk at least.
From Wikipedia however, it doesn't look particularly taxing. The views in google image search look nice though.
There's also one other mountain mentioned on tour pages - Mt Zuqualla - but I suspect it's basically a guided tour to a lookout, no trekking as such.