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Can't comment on the Princess as we did this cruise and land tour of the Rockies a couple of years ago with NCL. It was excellent and the weather was good - warm enough in the Rockies for shorts and T shirts - although as a previous respondent said you will need warm clothes of the Athabascar glacier. Lake Louise was still frozen in early June but there are other turquoise lakes. The weather in Alaska was also quite good, we were lucky and didn't have rain and it wasn't cold although sometimes we needed jackets. The lumberjack show (I think it was in Juneau) is good fun. So yes, early June is a good time - have a great cruise.
We did this in 2003. Weather was good, most people wore shorts, except on the Athabasca Glacier. When on the coach it is easy to take a jacket with you. Go for the De Luxe coach tour if you can it. It includes half board and we even had a couple of picnic lunches or snacks. There might not be many dining options other than the hotel on the cheaper land tours. Perhaps someone else can answer that.
Most calls off our boat were near the towns, which were n't large. We only did the tour and train up the Yukon, very interesting and worth the price. It was a wonderful holiday which we would love to do again, especially the land tour. We wished afterwards we had booked the land tour of Alaska as well.