@JaapScherphuis may already have said this in a comment, but frankly, I don't know most of the words used there, so I'm going to answer here in the affirmative.
There are two things to prove:
(1) A cell diagram (one of those rectangular things) generates a DAG that's planar
(2) A planar DAG of the type you're talking about corresponds to a cell diagram.
The first of these can be shown by a "contraction" argument, which I'll show visually; the mathematical details involve writing down homotopies/isotopies, all of which involve compressing $\Bbb R^2$ along vertical segments in not-very-interesting ways; a proof by pictures really tells the whole story:
In short: for each vertical segment draw a red line; for each "cell" draw a blue horizontal line mid-cell; this gives you a nice almost-graph-like structure. Now contract along vertical lines so that the red lines become points, and you have the DAG, nicely embedded in the plane.
Hence: an embedded cell diagram corresponds to an embedded DAG.
What about the other direction? Well, in trying to build one of your cell diagrams, we know that each vertex in the DAG corresponds to a red vertical line of some length, and each blue edge corresponds to a cell. But where do the horizontal edges in the cell-diagram come from?
Answer: the faces of the embedded DAG.
So here's an algorithm to transform a single-source, single-sink embedded DAG (I know you said "planar", but a planar graph is one that can be embedded, so I'm going to assume embedded), with the source at the left, the sink at the right, and all edges going from left to right (i.e., the vertices have been shifted so that there are no 'back-pointing' edges -- possible because ... it's a DAG).
I'm going to illustrate with the same DAG, because it means less drawing for me. So here's our starting DAG:
Step 1: count the number, $n$, of vertices in the DAG, and sketch faint grey vertical lines at $x$-coordinates $1, 2, 3, \ldots, n$. (The sketching is just to make it easy to see what's happening in the rest of the process: the vertical cell-edges will lie along these lines). In the example, $n = 6$, and we get this:
Step 2: Read across the top of the DAG from source to sink. In our case, this gives nodes $1, 4, 6$. In the diagram, add a horizontal line at the top, with short vertical lines descending from nodes $1, 4,$ and $6$:
Step 3: Do the same for the bottom edge of the dag, and add a horizontal line at the bottom, with ascending vertical lines at the nodes that the bottom source-to-sink path meets (in our case, $1, 2, 3, 6$):
Now we come to the interesting part, where we have to fill in other horizontal lines.
Step 4: For each region that touches the top source-to-sink path,
A. add a horizontal segment corresponding to the vertices of the region
B. Extend the current vertical segments to meet it.
C. Delete the upper edge(s) of the region, and any vertices left unconnected
C. For each remaining vertex of the region, start a new vertical segment downwards
Here's the result for the leftmost region (vertices $1,2,3,4$) and then the right-hand one (vertices $4,5,6$); each shows the enlarged cell-diagram and reduced DAG.
We continue this process; the next region to process is the small one in the middle with nodes $3,4,5$; removing this leaves vertex 4 isolated, so the red segment corresponding to $x = 4$ is finished, but red segments at $x = 3, 5$ continue downwards:
There's one more region ($3, 5, 6$) to process, which brings us to this:
and once there are no more regions, we simply connect all remaining vertical red segments to finish the cell diagram:
and we're done.
The proof that this works, now that you have the algorithm, should not be too tough for you (I hope). There's no doubt that it produces a cell-diagram-like figure. You do have to ask "are you sure that the remaining bottom vertical segments match the remaining vertical segments from above?", but I suppose you could skip drawing the bottom ones in the early steps, and finish by simply extending the remaining vertical segments all the way to the bottom.
Anyhow, I hope this is of some help to you.
One small correction: the solid green line from left to right, just below the top edge of the cell diagram, should really be drawn at two different levels: each time you process a new region, you should draw the corresponding green line a little lower. There may be some subtlety here -- I'm not completely certain. (I believe that whether you draw the right-hand part of that green line above the left OR below the left, you get the same DAG, although different cell diagrams. That may tell you that this isn't really the best possible representation...). But I wasn't willing to go back and redraw everything!