The closest theme parks to the Pop Century resort are Epcot and Disney's Hollywood Studios. These are both about 2 miles away depending on where you're measuring, a bit too long of a hike in my opinion. But even if you did want to walk, there would be no safe way to do this. There are no walking walking paths or pedestrian byways.
As for resorts, the outermost end of the Pop Century resort is only about one quarter of a mile from the Caribbean Beach Resort, but again, there is no safe way to walk there. You could use Disney transportation to get around, but if you did not want to rely solely on that, you would, in my opinion, need to rent a car.
A few other things to think about ... Although a car might be useful to you, you're going to want to think twice about using it on Christmas and New Year's Day as there is always the possibility on these dates that the parking lots may be closed for a time. For more information on this click
HERE.
Another thing to think about is only renting a car for the first part of your trip. You could rent and return from Alamo or National at the Dolphin hotel. By about the 5th of January, the crowds will have thinned quite a bit, perhaps making Disney transportation a more attractive alternative.
By the way, you can get a good sense of the layout of WDW and the distance and relationships between the parks and resorts by looking at GoogleEarth. You can even see the entire park system in 3D. You can learn more about this
HERE.