Hi, can anyone tell me if it is the capacitor I need to replace since the fan on the outside central air unit will not kick on by itself, but it will come on if you start it by hand? And if so how do I go about doing that?

Sounds like you know what is bad, Be sure disconnect AND Power are off. Replace with Identical Capacitor. Can go higher in voltage. Best to stay same Mfd for Capacitor.

It may not be the capacitor. The contactor that switches the outdoor fan motor on, may have a bad coil, or have damaged contacts (burned, pitted, broken).

If you don't know how to test a contactor (safely) using a multimeter, have a qualified service technician check it for you.

The brown wires from the fan motor go to the run capacitor, small oval and metallic device with at least two wire going to it. The info on it should say something like 5UF/370vac or 5mfd 370 vac. You need to buy the same size as the old one. TURN OFF ALL POWER GOING TO SYSTEM WHEN Getting INFO AND REPLACEING PART.