Can you rout plywood?
ApplePly is a brand name for an all-hardwood veneer plywood that is higher quality than your typical construction or cheap hardwood plywoods.  (You may want to confirm that your &quot;applewood&quot; plywood is truly ApplePly).  ApplePly should be similar in quality to baltic birch plywoods and like baltic birch it is not supposed to contain voids.  You can find these higher quality sheet goods at a large lumberyard or cabinet supplier.
A router will work fine on the edges of either ApplyPly or baltic birch.  The glue in the plywood may dull the router bit slightly more quickly than solid wood but if you are only doing one desktop this is not really a factor.
From an aesthetic standpoint, you will be able to see the veneer layers on the edges.  Depending on the piece of furniture this may be what you desire, or not.
additional info and clarifications based on the comments:
As several have mentioned it's a good idea to take several shallow passes rather than try to get it done in a single pass.  This will help give cleaner cut.  What 'shallow' means kind of depends on what the profile and thickness is, but start with about 1/16&quot; and increase from there. Use a carbide bit rather than a cheaper high speed steel one.  Make sure that you understand where the bearing of the bit will be riding, especially if you are profiling both sides.  You should test all of these by working on the scrap edges of the sheet (or other scrap of the same material) before you cut it to final size.  Secure the sheet with multiple clamps if at all possible.  This is for safety and it will also give you more control and a better quality cut.
The direction that you move the router when working is important.  If you are working on the outside edge of a plywood sheet with a handheld router the correct direction is counterclockwise- start on the right and push the router leftwards.  This keeps the cutter rotating against the wood- you will actually have to push the router where you want it to go.  If the router is pulling you along, you are making a climb cut which is useful in some situations but not really recommended for beginners because much more effort is required to maintain control of the router.