Fixation rate at neutral loci
First of all, the $\mu$ is not expected time for a mutation to occur and get fixed; it is the rate at which mutations are fixed in the population. The basic result states that if neutral mutations arise at a locus at rate $\mu$ within individuals, mutations at this locus will be fixed in the population at rate $\mu$ as well. 

The expected time for a given neutral mutation to fix after arising, given that it is not lost from the population, is approximately $4N_e$, where $N_e$ is the effective population size. Kimura and Ohta (1969) derive this result using a diffusion approximation; Kingman (1982) develops coalescent theory and in the process obtains what in my opinion is a more elegant derivation of the same approximation.

Now what is the probability that some mutation goes to fixation in the population as a function of time? Even though the fixation rate in the population is $\mu$, this does not mean that with probability 0.5 some mutation fixes within $1/\mu$ time units. That probability will depend on the distribution of fixation events in the population. Here's a good way to think about it. If fixation events were evenly spaced and occurred at rate $\mu$, then with probability 1 you would have a mutation within an interval of length $1/\mu$. If fixation events were highly clumped, then you would have a very low probability of having a fixation event within an interval of length $1/\mu$ -- but if you did have one event, you would probably have many. If fixation events are distributed as a Poisson process--which I suspect they would be in a purely neutral model--the probability of having a fixation event within a time interval $1/\mu$ would be given by a (negative) exponential distribution and thus the probability of having at least one fixation event in a time interval of length $1/\mu$ would be $1-\frac{1}{e}$.