Coded caching technique is an efficient approach to reduce the transmission load in networks and has been studied in heterogeneous network settings in recent years. In this paper, we consider a new widespread caching system called $(K_1,K_2,U,r,M,N)$ two-dimensional (2D) caching-aided ultra-dense network (UDN) with a server containing $N$ files, $K_1K_2$ cache nodes arranged neatly on a grid with $K_1$ rows and $K_2$ columns, and $U$ cache-less users randomly distributed around cache nodes. Each cache node can cache at most $M\leq N$ files and has a certain service region by Euclidean distance. The server connects to users through an error-free shared link and the users in the service region of a cache node can freely retrieve all cached contents of this cache node. We aim to design a coded caching scheme for 2D caching-aided UDN systems to reduce the transmission load in the worst case while meeting all possible users' demands. First, we divide all possible users into four classes according to their geographical locations. Then our first order optimal scheme is proposed based on the Maddah-Ali and Niesen scheme. Furthermore, by compressing the transmitted signals of our first scheme based on Maximum Distance Separable (MDS) code, we obtain an improved order optimal scheme with a smaller transmission load.