[This is just a recap/editing of comments that seem to give an answer]

A suggestion about the second statement : follow the gradient of $\phi=d^2_x∘f$, where $d_x$ is the distance to $x\in N$ with respect to a real analytic metric on N, the gradient being taken with respect to another such metric on M.

For this to work, one has first to rule out the possibility that $\phi$ has critical points arbitrarily near $f^{-1}(x)$, but not in it (note that each $z \in f^{-1}(x)$ is a critical point of $\phi$).
But the *curve selection lemma* (CSL) is precisely adapted to this situation. Namely, if $(z_n)$ is a sequence of critical points of $\phi$ in $f^{−1}(N∖x)$, with $f(z_n)\to x$, one may assume by properness of $f$ that $z_n\to z\in f^{−1}(x)$. Then the curve selection lemma (applied to the semi-analytic set $Z=Crit(\phi)\setminus f^{−1}(x)$, and the point $z$ in its closure) gives an analytic curve $\gamma:[0,\delta[\to M$ of critical points of $\phi$ with $\gamma(0)=z$ and $f(\gamma(t))\neq x$ for $t>0$.
But this is absurd : $t\mapsto \phi(\gamma(t))$ would have to be constant.

To conclude the argument, one may resort to Lojasiewicz inequality (itself a deep consequence of CSL) $|\mathrm{grad} \phi(z)| \geq c \phi(z)^\alpha$ if $\phi(z)<\epsilon$, for some $\alpha\in[0,1[$, $c>0$, $\epsilon >0$. This implies that the gradient trajectories have uniformly bounded lengths since they also are those of $\phi^{1-\alpha}$, which has gradient norm $\geq(1-\alpha)c$ on $\phi^{-1}(]0,\epsilon[)$.

This implies that the neighborhood $\phi^{-1}([0,\epsilon[)$ deformation retracts to $f^{-1}(x)=\phi^{-1}(0)$.

It must be said, however, that this is only a small fragment of the theory of (semi-,sub-) analytic sets developed by Lojasiewicz, Hironaka, etc (with previous work by Whitney and Thom) since the 60's, and that I am not competent to retrace here the story of its development, nor to give proper attributions. In particular, I don't know to what extent it relies on Hironaka's resolution of singularities.

Some references I've found are Milnor's 1969 *Singular points of complex hypersurfaces* where I first learned
CSL (in the semi-algebraic context), Denef and van den Dries 1988 *p-adic and Real Subanalytic Sets*,
Bierstone and Milman's 1988 *Semianalytic and subanalytic sets*, Kurdyka's 1998 *On gradients of functions definable in o-minimal structures*. Two papers by Lojasiewicz seem also very relevant (but not easily accessible).

curve selection lemmaadapted to this situation ? Namely, if $(z_n)$ is a sequence of critical points in $f^{-1}(U\setminus x)$, with $f(z_n)\to x$, one may assume by properness that $z_n\to z\in f^{-1}(x)$, and by selection lemma there is an analytic curve $\gamma:[0,\delta[\to M$ of critical points with $\gamma(0)=z$ and $f(\gamma(t))\neq x$ for $t>0$. It remains to rule this out... – BS. Dec 31 '10 at 15:47someneighborhood in $M$. But this surely results from sharp results on the structure of analytic sets (triangulability - in a "relative" situation), – BS. Dec 31 '10 at 16:13