you are going to WANT to bring your own coffee. I don't recall a speciality coffee place at AP.
ITA. I almost always prefer to make my own coffee in the room. It takes about five seconds to set up the machine, and five minutes to brew. It would take me at least that amount of time just to walk to AP. Then, you have to get the coffee, pay for it, and walk back. I'm the same way when I'm home, though. I hate hate hate to have to go out just for coffee. To me, it's more leisurely to lounge with my coffee.

But, YMMV. Some people just hate to make coffee and would rather go out.
It's nothing to pack some coffee and basket filters for the trip. Buy milk when you get there. Most times, we also purchase water. The taste of the tap water can leave something to be desired.
Now, depending on where you are located in relation to DTD, you could also go to Goofy's Candy Co. at the DTD Marketplace. They have coffee, cappucino, lattes etc. Last time there, we were in Congress Park, so one morning when DH was sleeping in, I strolled over to DTD to check it out (having never been there) and discovered that I could buy something other than Nescafe

It was maybe a 10 minute walk, if that.