It is because "arr" inside string do not bind to the input argument. You have to do it separatly.
a = {{1, 2}, {3, 4}};f[arr_] := ToExpression[ToString[arr] <> "[[1,1]]"]And now it works
But this type of coding can easily lead to errors and is fragile. It is better to find different approach for what you are trying to do.
takes in an array of unknown size
There is a command in Mathematica to find the size of list, it is called Dimensions and this way the called function can find the size of the input.