Can I replace just the knob of this outside spigot to fix this leak?
If you're lucky, with a frost proof fixture like this, you may be able to slightly tighten the nut that's holding in the packing or gasket behind the handle. You'll need to have two wrenches for this, one to hold the faucet in place, and a second to tighten the nut.

If this nut doesn't want to turn (don't force it, else you may damage it and then need to replace the entire faucet), then you'll need to replace the packing or gasket around the valve stem. It's a cheap repair, but requires the water is off to the fixture and that you are gentle to avoid damaging the soft metal.


shutoff the water
open the faucet to be sure it's turned off
remove the handle (usually held in with a screw)
remove the nut with two wrenches, one holding the faucet in place and a second to turn the nut
remove the valve stem by unscrewing it and pulling it out
get a replacement packing (looks like some black string you wrap around the stem a couple times) or gasket to match what's currently there
reverse the above steps to reassemble the faucet