Most people regard Nairobi as a place to get out of, as quickly as possible. It has acquired for itself the nickname Nairobbery, and not without cause. Some Africans we know don't spend more time in the city than they really have to.
However, many safaris arrive in Nairobi in the early morning and you have a day and overnight in the city before starting your safari the following morning. This is not always a bad idea, particularly if you are arriving from the States or another far-flung country. Sometimes there may be time in Nairobi before your return flight, or an onward flight to, for example the coast, Zanzibar, Seychelles or Mauritius.
I would always from choice start my safari right from the airport, but if it's not possible, there links at the bottom of this page are some suggestions for how to spend your time.
Other possibilities are:
The Museum: worth a few hours' visit. There are permanent early history and wildlife exhibits, and temporary exhibitions on various specialities.
Take a taxi from your hotel, (the dereva will normally wait for you, or come back at a time or by phoning) or if you really must, take it as part of a city tour, but then you will be time-limited.
The Bomas of Kenya: traditional-style dancing by professional dancers. I haven't visited this! However, I did catch a performance from the 'Bomas' troupe at The Carnivore Restaurant on our last safari, and the standard was very high.
NB: The Carnivore Restaurant is now closed.
If you have two nights to spend in Nairobi, you might like to visit The Ark, Treetops or Mountain Lodge, or do an overnight at one of the game parks or private reserves - Delamere Camp at Lake Elmenteita springs to mind.