It takes only some little algebra to get the result.
Using the fact that $X_t= X_s.e^{-\kappa(t-s)}+\theta.(1-e^{-\kappa(t-s)})+\int_s^t \sigma e^{-\kappa(t-u)}.dW_u$ equation (1),
$Y_t$ can be rewritten for any $s<t$ as :
$$Y_t=Y_s +\int_s^t X_udu$$
Now using equation (1) :
$$Y_t=Y_s + X_s.\int_s^t e^{-\kappa(u-s)}du + \theta.\int_s^t(1-e^{-\kappa(u-s)}) du + \sigma.\int_s^t\int_s^u e^{-\kappa(u-v)}dW_v.du $$
So now everything becomes clear once we use stochastic Fubini in the last integral (equation (2)):
$$Y_t=Y_s + X_s.g(s,t) + f(s,t) + \sigma.\int_s^t[\int_u^t e^{-\kappa(u-v)}du].dW_v $$
where $f$ and $g$ are suitable deterministic functions. As $X$ is known to be Markovian (classical result form solution of SDE equations gives this result see for example Protter), we have that the couple (X_t,Y_t) is Markovian using equation (2).
Best regards.