How should I touch up my exterior door's paint after changing hardware?
You could buy a sprayer for about $100 or less; I think a painter will charge you at least that much to spray it.

A sprayer such as this one or a wagner type airless sprayer should work Wagners are generally good for one or two uses unless you get a pro unit - ~$290.00 . I have had a couple of the Wagners , managed to get one to last through 4 rooms.

Here is how you can spray it.


Remove the door by removing the Hinge Pins.
Remove your hardware from the door.
Stand your door in a way that you can spray it while holding the
sprayer  horizontally - keeping the nozzle about 12 to 18 inches from
the surface of the door.


Hold the spray nozzle straight towards the door and walk along the door and spray (start from the top and work your way down - in case you create a run - you can fix it (wipe it) and continue) - remember keep the nozzle straight - no flexing of the wrist. 

Start spraying before you reach the door and finish spraying when you have passed the door. 

Continue this process until the door is completely painted. 

Let the door dry. 

Reinstall the door hardware and the door. You might need a helper as putting the hinge pins back in and holding the door may be difficult - putting shims under the door to hold it up and in place while installing the pins-  will make it easier to do it alone.

The best ambient temperature fro spray painting like this is around 70 degrees, if it is 90 degrees - move your nozzle a little closer to the door - 12 inches instead of 18 as the paint will tend to dry faster, if it is colder the issue is that of adhesion, your paint may not stick so well.