Finding Mixed-Strategy Subgame-Perfect Equilibrium
Consider the subgame following 1's choice of $Y$. This is simply a simultaneous move Battle of Sexes game. To compute the SPNE, you first need to find the Nash equilibrium of this subgame. 

It has 3 Nash equilibria: 2 pure and 1 mixed. The mixed strategy gives a value of $\frac{2}{3}$ to player 1 in this subgame. Now move up to the root node and compare whether player 1 should choose $Y$ or $N$. 
This would complete the computation of SPNE.