This may be recognised as the real part of a certain Fourier transform, which in turn is known to $\to 0$ as $n\to \infty$, by the Riemann-Lebesgue lemma.
From here i am interested in the main asymptotic behaviour for large $n$. However, I have not been able to make a good guess on how the base function $f$ effects the asymptotics.
For example, if $f(x)=e^{-ax}$ for $a>0$, then $\int_0^{\infty}f(x)\cos(nx)dx=\frac{a}{a^2+n^2}=O(n^{-2})$, for $n\to \infty$.
In addition, for the case $f(x)=e^{-ax^b}$, where $a,b>0$, we may use Mellins inversion formula and the residue theorem to find that $\int_0^{\infty}f(x)\cos(nx)dx=O(n^{-1-b})$.
I welcome any suggestions for any other special cases (e.g. $e^{-e^x}$ which is not solvable by the above methods ) or other general properties of the fourier transform.