You need to be careful about what you mean by "a general point". Usually, this means "a point in a certain Zariski open set". So in particular, this statement would say that on a surface of general type all rational and elliptic curves lie in a Zariski closed subset. This is Lang's conjecture, still open I believe (proofs were suggested about 15 yrs ago but then withdrawn).

EDIT: OK, so from the comments and the other answer it appears that it should be a "very general point of $X$", and the statement reduces to showing that if you have a morphism $f:Y\to T$ with irreducible $T$ and general fiber $Y_t$, and a finite dominant morphism $\pi:Y\to X$, then $Y_t$ is also of general type.

The basic reason for that is very simple: if $X$ is of general type then it has lots of pluricanonical forms. You can pull them back to $Y$ (they are differential forms, after all) and get lots of pluricanonical forms on $Y$. Then you can restrict them to $Y_t$ and get lots of pluricaninical forms on $Y_t$.

For a more precise answer, I suggest you look at old papers by Kawamata, Viehweg and Kollar, search for "additivity of Kodaira dimension". There is a whole sequence of $C_{n,m}$ conjectures about the Kodaira dimension of a fibration $Y$ in terms of Kodaira dimensions of $T$ and $Y_t$. Some of them are proved, some are still open.

(Note: general type means "maximal Kodaira dimension", i.e. equal to the dimension of the variety.)