Is there an "analytical" version of Tao's uncertainty principle?

One does not need Gaussians in the finite case, just take $f$ to be the indicator function of the interval $[-(n-1),n-1]\subset\mathbb F_p$. A simple computation gives $$ |\hat f(x)| = \frac1{\sqrt p} \frac{|\sin\pi(2n-1)x/p|}{|\sin\pi(x/p)|} < \frac1{2\sqrt p\|x/p\|}, $$ where $\|x/p\|$ is the distance to $x/p$ from the nearest integer. As a result, $$ \sum_{x=m}^{p-m} |\hat f(x)|^2 \le \frac p{2m} = \frac p{2mn} \sum_{x=0}^{p-1} |\hat f(x)|^2. $$ Taking, say, $m,n\approx\sqrt{\varepsilon^{-1}p}$, you have both $f$ and $\hat f$ concentrated on intervals of length about $2\sqrt{\varepsilon^{-1}p}$.

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A historical point worth clarification. The inequality $|{\rm supp}\, f|+|{\rm supp}\,\hat f|\ge p+1$ was in fact proved first by Andras Biro, who has contributed it as a problem to the 1998 Schweitzer competition (Problem 3). Bearing in mind that Tao's paper appeared in 2005, I think it would be most reasonable to call it the Biro-Tao inequality.

No. One just has to apply the standard example showing the classical uncertainty principle is sharp:

Let $f(a) = \sum_{n \in \mathbb Z} e^{- \pi ( a+pn)^2 / p}$. Then $\hat{f}$ is proportional to $f$.

But $1-\epsilon$ of its mass is contained in an interval of width something like $O ( \sqrt{ p \log (1/\epsilon)})$, which is much smaller than $\epsilon p$.

The counterexample to Biro–Tao uncertainty principle for a non-prime $p = k \times l$ is a vector $f$ given by $f(x) = \sqrt{k}$ if $x$ is a multiple of $k$ and $f(x) = 0$ otherwise. Then it is easy to see that $\hat{f}(x) = \sqrt{l}$ if $x$ is a multiple of $l$ and $\hat{f}(x) = 0$ otherwise. Taking $k = l$ large enough produces a counterexample to your claim: all of the mass of $f$ and $\hat{f}$ is contained in a set of cardinality $k = \sqrt{p}$.