If $p\in\mathbb{Q}[X]$, then the rational root theorem gives us possible integer roots of $p$. If $p\in\mathbb{R}[X]$, the theorem cannot be applied. Nevertheless, triangular inequality gives us lower and upper bound for all roots. If we define
$$p(x) = a_n x^n + \cdots a_1 x + a_0\text{,}$$ we notice that $$|p(x)|\,\geq\, |a_n||x^n| - \sum_{i = 0}^{n-1} |a_i||x^i|\;\geq\, |a_n||x^n| - \sum_{i = 0}^{n-1} A|x^i|\text{,} $$ where $A = \max_{0\leq i\leq n-1} |a_i|$. If $|x|>1$, it follows that $$|a_n||x^n|> An|x|^{n-1}\Rightarrow |p(x)| > 0\text{.}$$ Define $B = \max \{1, An/a_n\}$. Then $$|x|> B \Rightarrow p(x)\neq 0\text{.}$$
Thus, we obtain a finite number of possible integer roots.
Are there any considerably better estimates for the region where integer roots might lie?
No, in general there aren't any.