Hi there! We are just back from the PBH, and can tell you that we walked everywhere about 75% of the time. From reading these boards, I thought the PBH was in the boondocks (Easterbib, Universal!). The HRH is just a 5-7 minute walk from the PBH, and the HRH is pretty close to everything else.
I would give it a 12-15 minute walk to UO. Same to City Walk. Another 5 or so minutes to IOA. We were in the Villa Wing. It would be about the same if you are in the West Wing. Longer if you're in the East Wing.
Keep in mind that we are 2 adults, no kids, and we weren't there during any humid times. I'm sure the walk would seem a LOT longer if you're pushing a stroller, or you've got a lollygagging child or two, and it were 95 degrees.
We enjoyed the walk -- it's a very pretty one. We'd often walk "home," even after strolling the parks for hours.
If you don't want to walk, the boat will drop you off at City Walk. Boat transportation is frequent (there's never much of a wait at all).
Enjoy it!