Doubts regarding Central Limit Theorem

All,

I am studying CLT and I had a doubt.

I understand that sampling distribution has a mean same as population mean and std.dev as 6/sqrt(n) where n is sample size. Each point on X axis of Sampling dist is a mean of a random sample of sample size N. And by Z scores we can find the probability of a random sample having a particular mean.

I did few exercises on finding the probability. However, one particular question made me feel I still don't know CLT.

If you have taken a random sample with 121 observations from a population with 625 observations, mean=34 and standard deviation=4.1, what would be the standard deviation of that sample?

The answer to this exercise is 4.1/sqrt(121).

My doubt is that this formula applies to a sampling distribution. We are randomly taking samples which can take any value of mean and std.dev. How can we say that a std deviation of a random sample is same as that of sampling distribution. While I believe that more no of sample sets will have a mean same as that of Population but how can we say anything about std deviation. We can only confirm the standard deviation of a sampling distribution. I feel as if that answer to the above question means that we can assume sampling distribution as a distribution of one Random Sample (which I believe isn't correct).