The calculation of such probabilities has been studied extensively by communications engineers under the name $M$-ary orthogonal signaling where the model is that one of $M$ equal energy orthogonal signals being transmitted and the receiver attempting to decide which one was transmitted by examining the outputs of $M$ filters matched to the signals. Conditioned on the identity of the transmitted signal, the sample outputs of the matched filters are (conditionally) independent unit-variance normal random variables. The sample output of the filter matched to the signal transmitted is a $N(\mu,1)$ random variable while the outputs of all the other filters are $N(0,1)$ random variables.
The conditional probability of a correct decision (which in the present context is the event $E = \{X_0 > \max_n X_n\}$) conditioned on $X_0 = \alpha$ is $$P(E \mid X_0 = \alpha) = \prod_{i=1}^n P\{X_i < \alpha\} = \left[\Phi(\alpha)\right]^n$$ where $\Phi(\cdot)$ is the cumulative probability distribution of a standard normal random variable, and hence the unconditional probability is $$P(E) = \int_{-\infty}^{\infty}P(E \mid X_0 = \alpha) \phi(\alpha-\mu)\,\mathrm d\alpha = \int_{-\infty}^{\infty}\left[\Phi(\alpha)\right]^n \phi(\alpha-\mu)\,\mathrm d\alpha$$
Work in progress