Suppose a random variable $N \sim \operatorname{Pois}(\mu)$, with Gamma conjugate prior such that $\mu \sim \operatorname{Gamma}(\alpha, \beta)$. Then given a sequence of $n$ observations of counts $\mathbf{k} = (k_1, \dots, k_n)$ in the next $n$ units of time, then the way to derive the posterior distribution of $\mu|\mathbf{k}$ is by updating the prior so that:
$$\mu|\mathbf{k} \sim \operatorname{Gamma}(\alpha + \sum_{i=1}^n k_i, \beta + n)$$
Suppose instead of knowing the counts $\mathbf{k}$, I only know that the next success was observed in $t$ units of time. Is it then appropriate to update the prior so that:
$$\mu|\text{time to success} = t \sim \operatorname{Gamma}(\alpha + 1, \beta + t)$$
If this is appropriate, why does this hold and how does it relate to the previous update, and if it is not appropriate, how do you incorporate one observation of time of next success?
For example, I can see that when the time of next success is a small fraction of the unit time, and let $t_1, \dots, t_{k_1}$ denote the inter-arrival times of the first $k_1$ successes, then
$$\operatorname{Gamma}(\alpha + k_i, \beta + 1) \approx \operatorname{Gamma}(\alpha + k_i, \beta + \sum_{i=1}^{k_1} t_i)$$