The AP doesn't make you eligible for the Dining Plan; you'd have to gain eligibility either by purchasing a qualifying MYW package (including MORE admission tickets, which of course you don't necessarily need), or by booking a DVC reservation on DVC points.
The AP does qualify you for the Disney Dining Experience discount card. I believe the price is $65 and I found that if I was already a AP holder, I would have been better off, overall, going DDE instead of DDP. YMMV.