My new rabbit has been in a run for most of day on the lawn and eating grass is this safe?

As long as you haven't sprayed any insecticides or weed killer recently, this should be fine. I used to own a rabbit and I would let him roam the yard and graze and he lived for many years perfectly healthy. Just pay attention to when your city plans to spray for mosquitos, as any chemical that is used on or around the grass that your rabbit is eating can make him/her very sick.