I'll give a slightly uncertain answer, based somewhat on my recollection of conversations with Zagier a month ago about similar questions.

If we were to imitate Euler, we might consider $f(1)$ as
$$f(1) = \sum_{n \geq 1} a_n = \sum_{n \geq 1} a_n n^{-0} = L(f,0).$$
So the analytic continuation of the L-function suggests that $f(1)$ should be identified with the value of the L-function at zero. By the functional equation, this relates to the L-function at the right edge of the critical strip.

So, for a cusp form of weight two, arising from an elliptic curve $E$ over $Q$, the value $L(f,0)$ is related to $L(E,2)$. An interpretation of this L-value, conjectured by Zagier, was proven by Goncharov and Levin, in "Zagier's conjecture on $L(E,2)$", Invent. Math. 132 (1998).

As for the analytic question, you are considering the "value" of a cusp form $f$ on the real axis, which bounds the upper half-plane. Almost by definition, there is a Sato hyperfunction $f_{bdr}$ on the real axis, which describes this boundary behavior of the holomorphic function $f$ on the upper half-plane. I am not sure if the following is published, but I have the impression that there might be a preprint now or soon which proves the following result:

At every (positive? I don't recall) rational number $q$, the hyperfunction $f_{bdr}$ is $C^\infty$ at $q$. Its value at $1$ is $L(f,0)$ as described above.

I think that saying "a hyperfunction is $C^\infty$ at $q$" means that the hyperfunction can be expressed as the distributional derivative of a continuous function -- $f = g^{(k)}$ for some $k \geq 0$ -- and $g$ happens to be $C^\infty$ at $q$. But I'm not much of an analyst.

I think that the value $f(1)$ also exists as $\lim_{z \rightarrow 1} f(z)$ limit, if $z$ approaches $1$ via a geodesic in the upper half-plane.

I don't think you'll see Sha or the torsion directly, as these appear at the central value $L(f,1)$. On the other hand, I do think you'll find $L(f,-n)$ for all $n \geq 0$ (or equivalently, $L(f,2+n)$ ), by looking at the derivatives $f^{(n)}(1)$ of the boundary hyperfunction of $f$ at $1$.