So let me consider only the Gaussian case: Here also two possibilities - Stationary and non-Stationary.
Stationary Case
Let $\{x_t\}_{t=0}^{N}$ be an instance of the stochastic process $\{X_t\}$. Since, we are assuming that $X_t$ is stationary, let's also assume that we have a good model to forecast it.
Let $\mu_h \equiv \hat{X}_{t+h}$ be the estimate of $E[X_{t+h}|\{x_t\}_{t=0}^{N}]$; and $\sigma_h^2$ be the estimate of $Var(X_{t+h}|\{x_t\}_{t=0}^{N})$ that we have from the forecasting model.
A Side note: The standard error reported by forecasting model is not the $\sigma_h^2$ we need. The reported standard error is the standard deviation of $\mu_h$ (which is an estimate of actual mean). For details see this.
Coming back to the question. Since we have estimates of the mean and variance of $X_{t+h}$, we know the approximate asymptotic distribution of $X_{t+h}$.
So, $X_{t+h} \sim AN(\mu_h,\sigma_h^2)$. So $P(X_{N+h}>u)=1-P(X_{N+h}\leq u)=1-F(u|\mu_h,\sigma_h^2)$; where F is the normal CDF with $\mu_h,\sigma_h^2$ parameters.
Non-Stationary Case
This would follow the same concept but the result will be fairly ugly.
So assume that while $\{X_t\}$ is not stationary, $\{\Delta X_t\}$ is stationary.
Now, what we have from the forecasting model, instead, is the estimate of $E[\Delta X_{t+h}|\{\Delta x_t\}_{t=1}^{N}]$
See that, $X_{N+h}=\sum\limits_{j=1}^{h} (\Delta X_{N+j})+X_N$
So, $E[X_{N+h}]=\sum\limits_{j=1}^{h} (E[\Delta X_{N+j}])+x_N$ ................................(1)
$Var(X_{N+h})=Var(\sum\limits_{j=1}^{h} \Delta X_{N+j})$
$=\sum\limits_{i=1}^h \sum\limits_{j=1}^h Cov(\Delta X_{N+i},\Delta X_{N+j})$
$=\sum\limits_{i=1}^h Var(\Delta X_{N+i})+\sum\limits_{k=0}^{h-1} (k+1)\gamma_{h-k}$ ...........................(2)
where $\gamma_j \equiv Cov(\Delta X_{t},\Delta X_{t+j}) \,\,\,\,\forall t$ (this flows from stationarity of $\{\Delta X_t\}$.
Now let $\mu_{dh}$, $\sigma_{dh}^2$ and $\hat\gamma_j$ be the estimates of $E[\Delta X_{N+h}]$, $Var(\Delta X_{N+h})$ and $Cov(\Delta X_{t},\Delta X_{t+j})$, respectively, from the model.
So using (1):
$\mu_{h}=x_N+\sum\limits_{j=1}^h \mu_{dj}$
Similarly from (2),
$\sigma_{h}^2=\sum\limits_{j=1}^h \sigma_{dj}+\sum\limits_{k=0}^{h-1} (k+1)\hat\gamma_{h-k}\,\,\,\,\,$(for $h<N$)
So now we have estimates of the parameters of the asymptotic distribution of $X_{N+h}$ so as in case of stationarity above, we can use:
$X_{t+h} \sim AN(\mu_h,\sigma_h^2)$
Hope there aren't any mistakes.