Does there exist a positive irrational number α, such that for any positive integer n...

This is called a Beatty sequence. There will indeed always be a prime in the sequence. The bound for the OP's sequence (provided $\alpha >1$) is
$p\le L^{35-16ϵ}{\alpha }^{2(1-ϵ)}{p}_{m+l}^{1+ϵ}$
where $L=\log (2\alpha )$, $p_n$ denotes the numerator of the $n^{\text{th}}$ convergent to the regular continued fraction expansion of $\alpha$, and $m$ is the unique integer such that $p_m\le L^{16}{\alpha }^2<p_{m+1}$. $ϵ$ can be chosen arbitrarily small, but l depends on $ϵ$.