Given a irreducible and positive recurrent countable state Markov Chain with unique stationary distribution $\pi$, is the following true
$$\frac{1}{n}\sum_{i=1}^{n}X_i \to \sum j \pi_j$$
Or, to claim the above aperiodicity is also needed
What about when the process has uncountable state space ?
No, aperiodicity is not needed, only irreducibility (which is the statement that for all $x,y$ there exists an $n$ such that $P^n(x,y)>0$). In fact, even for a periodic or cyclic chain you get the result, that the Cesaro average of $P^n$ converges to a matrix in which every row equals $\pi$. Your result then follows from this.
The proof is based on the fact that for a period $d$, you can interpret $P^d$ as the transition matrix of a Markov chain with $d$ separate ergodic sets that are all aperiodic. Hence, $P^{kd+l}$, for $l\in\{0,...,d-1\}$, has $d$ converging subsequences. If a sequence has converging subsequences, it is Cesaro summable.
For the details I can recommend Theorem 5.1.4 in Kemeny and Snell's "Finite Markov Chains".