Is it possible to find a distribution of $X(t)$ for a fixed t by looking at a single sample path of X?
I'm kinda lost in the strong assumption, that it is not possible but then I remember that for time series analysis something related to likelihood method implies working with distributions and we have just 1 path there. Is there a way to prove the possibility or impossibility of such thing?
The concept of stationarity and ergodicity (see the wikipedia pages: Ergodicity, Stationary ergodic process) are used to infer several characteristics of a stochastic process by observing only a single sample path of $X$. For example you can compute the moments of the process given a path ${x_k}$ as follows $$ E[X_t^k] = \lim_{M\to \infty} \frac{1}{M}\sum_{t=1}^M x_t^k $$
In case that the distribution is completely characterized by several or all of the moments, then (in this case) it is in principle possible to characterize the distribution of $X$ using the given path.