Yes, plenty! But this is subtle since it's hard to define what an ultrafilter is in constructive settings.
An ultrafilter $\mathcal{U}$ on $\mathbb{N}$, in the classical sense, easily allows the Limited Principle of Omniscience (LPO). That principle says that if $\alpha(n)$ is decidable (either true or false for every $n$) then either $\forall n \alpha(n)$ or $\exists n \lnot \alpha(n)$. Given an ultrafiler $\mathcal{U}$, just check whether $\{n \in \mathbb{N} : (\exists m < n)\alpha(m)\} \in \mathcal{U}$$\{n \in \mathbb{N} : (\exists m < n)\lnot\alpha(m)\} \in \mathcal{U}$ to decide this.
Note that I needed the decidability of $\alpha(n)$ to make this work. It is not true in a constructive setting that every reasonable $\alpha$ is so decidable, so an ultrafilter might not help immediately. However, in most constructive settings, primitive recursive $\alpha(n)$ are decidable and this is already nontrivial. By induction, every arithmetical statement is decidable in this way given an ultrafilter $\mathcal{U}$. On the other hand, nothing else is since an ultrafilter predicate over second-order logic is conservative over arithmetic comprehension.