Well, I can tell you how to get to a couple of different Publix, but I don't know that they are the closest to Pop Century.
Go toward DTD. Keep going until you are facing The Crossroads. Turn left on CR-535. Just keep going straight and you will find yourself on Apopka-Vineland (CR-536). You will have about a ten minute drive. You will find a Publix on the left side of the road just before your reach Conroy-Windermere Rd.
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Go toward DTD. Keep going until you are facing The Crossroads. Turn left on CR-535. At the next light (you'll see an Olive Garden on the right), turn left. CR-535 actually turns to the left. Follow CR-535. It will be about a ten minute drive. Along the way, you will pass Disney's Distribution Center, Reams Road and Chase Road. Keep going until you see Summerport. There will be a Publix in the little shopping area.