The algorithm for constructing the AR-quiver of any orientation of $A_n$ is the same as the algorithm for constructing the AR-quiver of the orientation you describe. Start out by figuring out all the irreducible maps between the indecomposable projectives. Then write out the cokernels, and then the cokernels of the new maps, and so on until you get to all of the indecomposable injectives.
For example, consider the $A_3$ of the form $1\xleftarrow{} 2\xrightarrow{} 3$. You have two simple projectives $P_1$ and $P_3$ and the non-simple $P_2=(k\xleftarrow{}k\xrightarrow{} k)$. Both $P_1$ and $P_3$ map into $P_2$ with respective cokernels $I_3=(0\xleftarrow{} K\xrightarrow{k\xrightarrow{} K)$ k)$and$I_1=(K\xleftarrow{} K\xrightarrow{I_1=(k\xleftarrow{} k\xrightarrow{} 0)$. The cokernel of$P_2\to I_1\oplus I_3$is the simple injective$I_2$. So the AR-quiver is like a fish swimming to the right. 2 Added example The algorithm for constructing the AR-quiver of any orientation of$A_n$is the same as the algorithm for constructing the AR-quiver of the orientation you describe. Start out by figuring out all the irreducible maps between the indecomposable projectives. Then write out the cokernels, and then the cokernels of the new maps, and so on until you get to all of the indecomposable injectives. For example, consider the$A_3$of the form$1\xleftarrow{} 2\xrightarrow{} 3$. You have two simple projectives$P_1$and$P_3$and the non-simple$P_2=(k\xleftarrow{}k\xrightarrow{} k)$. Both$P_1$and$P_3$map into$P_2$with respective cokernels$I_3=(0\xleftarrow{} K\xrightarrow{} K)$and$I_1=(K\xleftarrow{} K\xrightarrow{} 0)$. The cokernel of$P_2\to I_1\oplus I_3$is the simple injective$I_2$. So the AR-quiver is like a fish swimming to the right. 1 The algorithm for constructing the AR-quiver of any orientation of$A_n\$ is the same as the algorithm for constructing the AR-quiver of the orientation you describe. Start out by figuring out all the maps between the indecomposable projectives. Then write out the cokernels, and then the cokernels of the new maps, and so on until you get to all of the indecomposable injectives.