I want to go for a run, can I leave my AirPod Case at home and just take one airpod with me

There is no limitation. The case is only required to charge and pair.

Once that is done, the case is not required to be close by. It can be 100 miles away, and the Airpods will still work with the paired iPhone.

The case has absolutely nothing to do with how far away they work, the case only charges and resets/pairs them.

If you own them cant you try it for yourself? Seems like a very easy and quick way to get the exact answer you are looking for.