Is that going to be the coop or a fenced run? and the door is going between the post and the shed, right?
Bracing the door so it doesn't sag isn't that big of deal, but you will have to run the brace from the bottom on the hinge side of the door, to the top corner away from the hinges. This way the weight of the door is held by the crossbrace and rests on the bottom hinge corner of the door. And the hinge should be located close to the bottom to support the weight of the cross brace.
One thing to remember is that a 4 foot screen door will have some "wiggle" to it and a predator (insert "coon" here) will have a good chance of pulling the bottom of the door away from the door frame and squeezing in. You could always put a latch at the top and bottom to prevent that from happening though.
If you wanted to make it a little narrower, you can just use 2x4 nailed in place between your base 4x4 and a 2x4 along the top. You really wouldn't have to use another 4x4 post and bracket between the existing 4x4 postand edge, since you don't need the strength that type of construction gives.
If you are in a snow area, and are putting a solid roof on the thing ,then a 4 foot spacing will be too much
without additional support for your roof and snow load.
EDITED TO ADD: I like FoxKeeper's idea also.