there are 4 sub games in the game.

The first/second game is with player 1 where he has to choose between (3,1) and (1,3) hence he will choose (3,1)

The second/first game is again with player 1 where he has to choose between (1,3) and (3,1) so he will choose (3,1)

Now the third game is where player 2 has to choose between (1,2) , (3,1) and ( 3,1)

for player 2, the outcome of 2 is higher than that of 1, hence he will choose (1,2)

Now the fourth game is where player 1 has to choose between (2,1) and (1,2)

For player 1, 2>1 and hence he will choose (2,1)

and (2,1) is the subgame perfect nash equilibrium

please see the attached file.