If you have four days, don't stay in Nairobi. Unless you have already done a proper safari somewhere (Serengeti, southern Africa, etc.), you MUST go to the Masai Mara. Even if you have done so, it may still be worth it.
If you do not want to go to the Masai Mara, then:
- You can see Amboseli (many elephants, Mount Kilimanjaro) can be done as a day trip, or as a 24 hour trip, departing mid-day, see the park in the afternoon and following morning, return mid-day.
- You can see Lake Nakuru (flamingoes) as a day trip from Nairobi. Or, combine it with Mount Kenya (great Fairmont there on William Holden's estate) and it might be a day and a half or so.
If you are committed to staying in/near Nairobi, there's the Nairobi game park just outside of town, the Karen Blixen house, and a few animal sanctuaries. Together that's the better part of a day.
In Nairobi proper, there are a few museums, the National Archives has some interesting old photos, the site of the former US Embassy is interesting if that's relevant to you, etc. Together these are also the better part of a day.