Yeah, I'd skip Ikebukuro. Not because it's unsafe, but because it's boring.

If by the "Tokyo Furi Kippu," you mean the Y1580 ticket that covers all subways and JR Tokyo, that is a waste of money. It's really hard to spend Y1500 in a day on transport plus I don't think it covers either the monorail or Keikyu line that is the first leg leaving Haneda.

The special JR tickets that include the monorail will not be available at Haneda before 11am.