I would not use an unsupervised method of outdoor restraint with a new puppy, i.e. a wire run, stake in the ground, etc. Your puppy will be too vulnerable to predators, e.g. foxes, coyotes, various raptors, free roaming dogs, etc.
An outdoor fenced in run is fine. Something like 6 x 20 ft or more. However until your pup is 12 or 15 pounds it needs some sort of overhead cover for protection from raptors. Since you need to provide shade anyway, some sort of tarp or other covering held on with bungees will work.
That said, an outdoor run will NOT help with housebreaking. That would be the major behavioral issue to consider.
The bottom line is crating and tethering (to you) until housebroken is the way to go. Unsupervised time in a large space (indoor exercise "ex" pen, outdoor run, etc.) will generally make housebreaking much more of a chore.