Complicated utility function
The demand functions you have derived are correct.
The Marshall demands are given as
$$ x^\star(p_x,p_y,M) = \sqrt{\frac{M}{p_y}} \\[8pt]
y^\star(p_x,p_y,M)  = \frac{M - p_x \sqrt{\frac{M}{p_y}}}{p_y},$$
where $M$ is income and $p_x$ is pizza price and $p_y$ is the apple price.
To answer the question of how the agent responds to change in pizza price it would be standard to find derivatives and/or elasticities of demand with respect to the price $p_x$ for pizza.
The derivative of pizza demand with respect to own-price is given as
$$ \frac{\partial x^\star(p_x,p_y,M)} {\partial p_x} = 0, $$
which implies that the elasticity is
$$El_{p_x}x^\star(p_x,p_y,M) = \frac{p_x}{x^\star(p_x,p_y,M)}\frac{\partial x^\star(p_x,p_y,M)} {\partial p_x}  = 0,$$
so you can conclude that demand is perfectly inelastic. When the price increase the consumer simply keeps demand constant and substitutes away from apples. This is manifest from the fact that
$$ \frac{\partial y^\star(p_x,p_y,M)} {\partial p_x} = - \sqrt{\frac{M}{p_y^3}}, $$
which is clearly negative and by implication the cross-price elasticity $El_{p_x}y^\star(p_x,p_y,M) &lt;0$.
This is &quot;unusual&quot; if you expected apples and pizza's to be substitutes with the demand of one increasing when the cross-price increase.