This is just 10 kms or 45 mins from the city centre but I've seen so many animals up close to me. I was lucky enough to see 10 Lions including 2 cubs apart from many other animals like Zebras, Buffaloes, Giraffes, Rhinos, Monkeys, Ostrich and others. The animals are used to seeing human beings & are normally visible from roads.
I had to travel in a Corolla (taxi used in city) since their safari vans were already booked.
TIP: Please reserve their vehicle in advance by calling them a couple of days in advance or go through a travel agent. The best thing would be their safari vehicles.