Proof that utility is nonincreasing in prices
That follows $v(p_1,y)$ being the highest utility one can get from a bundle affordable given prices $p_1$ and income $y$. Let $x_1$ be such a utility maximizing bundle. Then $p_1\cdot x_1\le y$ and $u(x_1)\ge u(x)$ for all $x$ such that $p_1\cdot x\le y$. In particular, $u(x_1)\ge u(x_0)$ since $p_1\cdot x_0\le y$ (this is what the argument you have shows.) Therefore,
$$v(p_1,y)=u(x_1)\geq u(x_0)=v(p_0,y).$$
Intuitively, if prices are lower, one can buy from a richer set of options, and with more options one can always get at least as high utility.