As Donna pointed out, the North Rim closes for the winter, and its opening date in the spring depends on weather conditions that cannot be predicted in advance, but mid-May is typical. This year there is an additional consideration. US-89 is closed between Page and Bitter Springs AZ. Repairs are still in the planning stages, but it is clear that work will take so long that the 'quick' solution is to pave an entire BIA highway through the Navajo Nation. That temporary fix was just approved on April 12th. What all this means is that your best options for getting from Bryce/Zion to the South Rim will either be through Las Vegas, Kingman and Williams; or down Alt-US-89 from Kanab through Fredonia and Jacob Lake to Bitter Springs where you would reconnect with US-89 south to Cameron and the entrance to the South Rim. But Jacob Lake is where the road into the North Rim (AZ-67) starts from. So you're going to have to do a little homework now to figure out what each of those options means to you in terms of miles and time, and then some more homework just before you leave to determine what's actually open in terms of both roads and park sections.

AZBuck