Estimate probabilities from its moments I want to estimate probability $Pr(X \leq a)$, where $X$ is a continuous random variable and $a$ is given, only based on some moments of $X$ (e.g., the first four moments, but without knowing its distribution type). 

The entire sequence of moments of a random variable $m_k = \mathbb{E}(X^k)$ determines the distribution function of $X$ uniquely, provided that $\sum_{k=0}^\infty \frac{m_k}{k!} t^k$ converges for all $t$ in an open neighborhood of $t=0$. See this.
If you have two such sequences, which coincide up to order $r$, but differ afterwards, these sequences correspond to different distributions. 
You may, however, ask to approximate the distribution function $F_X(x) = \mathbb{P}(X \leq x)$ given the values of the low order moments, if some assumptions on the nature of the distribution is made.  See method of moments estimation, for example. 
Knowledge of moments, determines an upper bound on the tail of the distribution function. See Chernoff bound, and Chebyshev inequality.
You may also find Pearson distribution, determined by first 4 moments, useful
