Let $f(x)$ be a continuous probability density function (PDF). What is the best method to estimate $f(a)$ for a given single point $a\in\mathbb{R}$ given i.i.d samples of the PDF (i.e. $x_i \sim f(x), i=1,\cdots, n$)?
Should I estimate the entire density, for example using kernel density estimation, to find an estimator at a single point $f(a)$?
Is it possible to perform the estimation without estimating the entire PDF?
Do you imply that you do not have an analytical expression of your PDF? If you only have the data points $x_i$ then I can only see that you have to somehow estimate the PDF from those and then evaluate this estimation at $\alpha$. As you said using kernel density estimation is a way to solve your problem, though different smoothing kernels can be used. I would also suggest reading this (most probably you have already done so)