Solution:

Since $1183 = 7 \cdot 169 = 7 \cdot 13^2,$ we have $\sqrt{1183} = 13\sqrt7.$ Square $\sqrt x + \sqrt y = 13\sqrt7$ to get $x + y + 2\sqrt{xy} = 1183.$ So $\sqrt{xy}$ is rational, which forces each of $x, y$ to be $7$ times a perfect square: $x = 7a^2,$ $y = 7b^2$ with $a + b = 13.$ Now $x + y = 7(a^2 + b^2),$ smallest when $a$ and $b$ are as equal as we can make them. Take $a = 6,$ $b = 7$ for $7(36 + 49) = 595.$ Thus, B is the correct answer.