The crab cakes can be formed and chilled a day ahead, and can be fried up to an hour ahead; reheat them in a 250-degree oven on a baking sheet lined with a wire rack just until warmed through, 8 to 10 minutes.
In a large bowl, whisk together egg, mayonnaise, Old Bay, and 1/2 teaspoon salt. Fold in crabmeat to evenly coat (do not overmix). Generously dust a parchment-lined rimmed baking sheet with cornmeal. Divide crab mixture evenly into 12 portions, forming each into a 2-inch-diameter, 1-inch-thick patty. Transfer to prepared baking sheet. Dust tops of crab cakes generously with cornmeal and chill in refrigerator until set, at least 1 hour and up to 1 day.
Stir together celery, cucumber, juice of one lime half, and a pinch of salt; set aside.
Heat 2 tablespoons oil in a large nonstick skillet over medium-high. When oil shimmers, add half of crab cakes and reduce heat to medium. Cook, undisturbed, until a golden-brown crust forms on bottoms of cakes, 4 to 5 minutes. Gently flip and cook until golden brown on other sides, 4 to 5 minutes more. Transfer to a platter and loosely cover with foil to keep warm. Wipe skillet clean and fry remaining cakes in remaining 2 tablespoons oil.
Squeeze remaining lime half over crab cakes. Serve warm, topped with celery mixture.