Let $X$ be a random variable and $f,g$ be two nice functions. In this context, I view $g$ as a good approximation to $f$.
I am interested in obtaining bounds for the following probability which measures the discrepancy in the distributions of $f(x)$ and $g(x)$. Can I bound $|P(f(x) <a) - P(g(x)<a)|$ by a quantity of the form $P(|f(x)-g(x)|>\epsilon)$? If I'm able to do so, I could then use inequalities that bound the discrepancy between $f$ and $g$. It doesn't really have to be $P(|f(x)-g(x)|>\epsilon)$ but any probability of the form $P(|f(x)-g(x)| \in B)$ for some set $B$ will do.
The closest inequality to this that I've seen comes from Wikipedia: see this link. But it is not quite what I need.
You don't specify the quantifiers on $a,\varepsilon$, and not even whether you wish to get upper or lower bounds of the first quantity in terms of the second. So this answer is going to stay very high level.
You most likely will not get any meaningful bound between the two quantities, since they are dealing with qualitatively different things; observe that the quantity $$ \lvert \mathbb{P}\{ f(X) < a \} - \mathbb{P}\{ g(X) < a \} \rvert \tag{1} $$ says something about the distributions of $f(X)$, $g(X)$ (for instance, its value would be unchanged when replacing $g(X)$ by $g(Y)$, for $Y$ an independent copy of $X$). The second, $$ \mathbb{P}\{ \lvert f(X) - g(X) \rvert > \varepsilon \} \tag{2} $$ says something about the random variables $f(X),g(X)$ themselves (not just their distribution).
As an example: take $f$ to be the identity $f(x)=x$, $g$ to be the negation $g(x)=-x$, and let $X$ be a Rademacher random variable (i.e., uniform on $\{-1,1\}$). (For a similar example, you could take $X$ to be a standard Gaussian.)
Then, for all $a\in\mathbb{R}$, $$ \lvert \mathbb{P}\{ f(X) < a \} - \mathbb{P}\{ g(X) < a \} \rvert = 0 $$ while $$ \mathbb{P}\{ \lvert f(X) - g(X) \rvert > \varepsilon \} = 1 $$ for every $\varepsilon < 2$.