If you have a mainly lower body program, like running, and you rest 3 days a week, will training arms on those days have an effect on the running?
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As long as you eat sufficient calories, your running workouts and push-ups will not interfere with each other.
The only effect I have seen is sometimes my arms or shoulders will get sore while I run. This all depends on how hard you go with both running and arm work.
I don't think working out your arms will HELP running in any way if that's part of your question whether it have an "effect". Although if you do find your arms ever getting tired during runs, strengthening them will help in the long run (no pun intended...well maybe)
All recovery is shared
All training takes a toll from all other training. This can be minimized by working separate parts of the body or separate metabolic pathways, but in the end your ability to recover from any workout is shared across all workouts, all you eat, all your stress, all your sleep.
Running and arms work sounds fine
That said, if you're used to running three times a week and you add in some push-ups and pull-ups, then you shouldn't experience a significant decrease in your recovery from either. Make sure that you have at least one day a week of complete rest--no running, no arm work, just taking it easy with a stroll or massage or nothing at all.
If you were doing an extensive bodybuilding routine (with many exercises, heavy weights, and many sets to failure), it might be a problem. If you're new to both modes of exercise, recovery would be affected, but you'd do fine. I'd be more concerned with getting an adequate balance of upper-body movements: be sure to mix pushes and pulls, strength work with endurance work, and so on.