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 equally likely
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 $C = \{X_0 > \max_i X_i\}$) conditioned
on $X_0 = \alpha$ is
$$P(C \mid X_0 = \alpha) = \prod_{i=1}^n P\{X_i < \alpha \mid X_0 = \alpha\}
= \left[\Phi(\alpha)\right]^n$$
where $\Phi(\cdot)$ is the cumulative probability distribution
of a standard normal random variable, and hence the unconditional
probability is
$$P(C) = \int_{-\infty}^{\infty}P(C \mid X_0 = \alpha)
\phi(\alpha-\mu)\,\mathrm d\alpha
= \int_{-\infty}^{\infty}\left[\Phi(\alpha)\right]^n
\phi(\alpha-\mu)\,\mathrm d\alpha$$
where $\phi(\cdot)$ is the standard normal density function.
There is no closed-form expression for the value of this
integral which must be evaluated numerically.
Engineers are also interested in the complementary event -- that
the decision is in error -- but do not like to compute this as
$$P\{X_0 < \max_i X_i\} = P(E) = 1-P(C)$$
because this requires very careful evaluation of the integral for $P(C)$
to an accuracy of many significant digits, and such evaluation is both
difficult and time-consuming. Instead, the integral for
$1-P(C)$ can be integrated by parts to get
$$P\{X_0 < \max_i X_i\} =
\int_{-\infty}^{\infty} n \left[\Phi(\alpha)\right]^{n-1}\phi(\alpha)
\Phi(\alpha - \mu)\,\mathrm d\alpha.$$
This integral is more easy to evaluate numerically,
and its value as a function of $\mu$ is graphed and
tabulated (though unfortunately only for $n \leq 20$)
in Chapter 5 of Telecommunication Systems
Engineering by Lindsey and Simon, Prentice-Hall 1973,
Dover Press 1991.
Alternatively, engineers use the union bound or Bonferroni inequality
$$\begin{align*}
P\{X_0 < \max_i X_i\} &= P\left\{(X_0 < X_1)\cup (X_0 < X_2) \cup \cdots
\cup (X_0 < X_n)\right\}\\
&\leq \sum_{i=1}^{n}P\{X_0 < X_i\}\\
&= nQ\left(\frac{\mu}{\sqrt{2}}\right)
\end{align*}$$
where $Q(x) = 1-\Phi(x)$ is the complementary cumulative normal
distribution function.
From the union bound, we see that the desired value $0.01$ for
$P\{X_0 < \max_i X_i\}$ is bounded above by $60\cdot Q(\mu/\sqrt{2})$
which bound has value $0.01$ at $\mu = 5.09\ldots$. This is
slightly larger than the more exact value $\mu = 4.919\ldots$
obtained by @whuber by numerical integration.
More discussion and details about $M$-ary orthogonal signaling
can be found on pp. 161-179 of my
lecture notes for a class on communication systems.
