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If $X$ is normal, then the Iitaka fibration theorem implies that $L$ is big if and only if the rational map

$\phi_m \colon X \dashrightarrow \mathbb{P}H^0(X, L^{\otimes m})$

is birational onto its image for some $m >0$, see [Lazarsfeld, Positivity in Algebraic Geometry I, p. 139].

This

I guess this is the "embedding theorem for big lines line bundles" you are looking for.

Regarding your last question, since $L$ is ample by assumption and $f$ is an isomorphism outside its exceptional locus $E$, some power of $f^*L$ will surely give an embedding of $X \setminus E$.
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If $X$ is normal, then the Iitaka fibration theorem implies that $L$ is big if and only if the rational map

$\phi_m \colon X \dashrightarrow \mathbb{P}H^0(X, L^{\otimes m})$

is birational onto its image for some $m >0$, see [Lazarsfeld, Positivity in Algebraic Geometry I, p. 139].

This is the "embedding theorem for big lines bundles" you are looking for.

Regarding your last question, the answer is "yes", since $L$ is ample by assumption and $f$ is an isomorphism outside its exceptional locus $E$, some power of $f^*L$ will surely give an embedding of $X \setminus E$.
show/hide this revision's text 1

If $X$ is normal, then the Iitaka fibration theorem implies that $L$ is big if and only if the rational map

$\phi_m \colon X \dashrightarrow \mathbb{P}H^0(X, L^{\otimes m})$

is birational onto its image for some $m >0$, see [Lazarsfeld, Positivity in Algebraic Geometry I, p. 139].

This is the "embedding theorem for big lines bundles" you are looking for.

Regarding your last question, the answer is "yes", since $L$ is ample by assumption and $f$ is an isomorphism outside its exceptional locus.