Several good ideas here so far, but I'll add another.
First, be sure to set the container view (your blue rectangle) to clip children, using the Clip Subviews checkbox in Interface Builder.
Second, add a set of child views with images of each numeral for each digit to be presented (4 * 10 = 40 child views in your case). Use tags in IB so you can easily access each one. Set the initial bounds to just below your bottom margin.
Call UIView animateWithDuration. In the animations block, set the new digit view's frame to appear in the clipped parent view's bounds, and set the old digit view's frame to just above the container's bounds. Since both view's frame changes are animated in the same block, this will create the effect of the new digit sliding up into place as the old digit slides out the top.
In the completion block, set the old digit's frame back to the original position below the container view.
With this approach, you can play with the duration and the timing curves, so that you cam emulate the pause that occurs with each digit fully displayed between transitions.
A similar approach can also be used with CALayer, or with sprites, but UIViews are lightweight, perform well, and easiest to assemble in Interface Builder.