Whoever is in this hospital is probably going to be rather important. There were numerous shots of it in the first trailer, including another one of Frank Castle there, so he is clearly after a very specific target. You would imagine that's a patient rather than a doctor, so it suggests that he has hospitalised someone, and is arriving here to try and finish the job. Based on what we've seen from Matt as well, we could expect some sort of Punisher/Daredevil showdown to occur at this hospital, which is shaping up to be a key location in Season 2 (kind of says it all about the kind of body count we can expect).