It would be easier to start with cube, as there is only 5 3-digit cubes (i.e. 125,216,343,512,729).
Note that 1 Across and 2 Down is cube.
There is two possible solution pair: 125 with 512 or 512 with 216.
However, 125 with 512 is not possible because "3 Across is square" and it would not end with 2.

1 Down start with 5. It must be either 529 (23 square) or 576 (24 square).
As we can find that there is no square start with 9 and end with 6, 1 Down and 3 Across must be 576 and 676 respectively.

The required sum= 576+676+512+216= 1980