This is as for north as you can go and not get wet, 330 miles north of the arctic circle.
As for a cruse ship schedule, I don't think so, the one person I talked to complained they promised a lot like a stop at little Diomede, and something else, and they didn't deliver, but they did go north to the pack ice. This would be a difficult stop for a cruse ship since sometimes the wind has the pack ice jammed ashore most the summer.
Yesterday it was snowing, but none was sticking.
Looks like Arctic cruise plies the legendary northwest passage, but no one lists Barrow as a port of call.http://www.polarcruises.com/arctic/destinations