Given a polynomial $P(x)$, is it possible to estimate/lower bound/upper bound the value of $P(k)$ for some $k \in \mathbb{N}$ if we know $P(1)$?
We can also assume $P(x)$ has only natural coefficients and is only taken over $x \in \mathbb{N}$ if that helps.
No if you assume that the coefficients can be in $\mathbb{Z}$, as you can just consider for example the family $$P_n(x)=(x-1)^n$$ You clearly have no control whatsoever on the growth rate, but all share the same value $$P_n(1)=0$$
In the case of positive integers, you too cannot find any upper bound, as the family of monomials $$P_n(x)=x^n$$ have similar properties as the one we said above, but for any polynomial $P(x)=\sum_{i=1}^na_ix^i$with positive coefficiens, it holds that $$P(x)\ge P(1),\quad\forall x\ge 1$$ because $x^i\ge 1 $ if $x\ge 1$ and so $$\sum_{i=1}^na_ix^i\ge \sum_{i=1}^na_i(1)^i=\sum_{i=1}^na_i=P(1)$$ Where the positivity of the coefficients is needed to pass from $x^i\ge 1$ to $a_ix^i\ge a_i(1)^i$.