As per Central tendency   ( x̅ - E(x̅))/ Std(x̅)  follows standard normal distribution with mean 0 and standard deviation of 1

i.e  ( x̅ - E(x̅))/ Std(x̅) follows N(0,1) or equivalently x̅ follows Normal with mean E(x̅) and variance var(x̅).

Note that E(x̅) = θ and as Var(x) = θ for a Poisson distribution 
Var(x̅) = θ/n which basically mean x̅ follows normal distribution with mean θ and variance θ/n. 





Hope this clarifies your answer