Find equilibrium price using excess demand function
Let $m_i$ be the market value of the endowment of player $i\in\{1,2\}$.
In optimum player $i$ spends everything such that $p_x x_i + p_y y_i =m_i$.
And because of the preferences in optimum also $x_1 = 4 y_1$ and $x_2=y_2$ holds.
You use that to get 2's demand as $x^*_2 = \frac{m_2}{p_1+p_2}=y^*_2$ and a similar bur slightly different formula for $4 y^*_1 = x^*_1$.
The excess demand would be this demand minus the endowment. The sum of excess demands is zero in equilibrium.
In equilibrium, $x_1^*+x_2^* =2+4$ and $y_1^*+y_2^* =2+4$, which you can both use to calculate an equilibrium price ratio. Any price combination with that ratio clears the market.