Expression of a gene is inherently random, leading to variability in a protein's level across a population of cells with same genetic information and environment. Another consequence of this is the cell-to-cell variability in the time at which a certain protein level is achieved inside individual cells. In this work, we model such times using the first-passage time (FPT) framework. Gene expression is modeled in translation bursts wherein each mRNA molecule arrives as per a Poisson process, produces a geometrically distributed burst of protein molecules and degrades instantaneously. Also, the proteins are assumed to degrade as well. The FPT probability density function and statistical moments are determined for this model. In addition, the effects of change in model parameters (transcription rate, mean translation burst size, FPT threshold) on the mean and noise (quantified as the coefficient of variation squared) of FPT are studied. Our analysis shows that the mean FPT increases by increasing the FPT threshold or decreasing the protein production by a lower mean burst size or a lower transcription rate. The noise properties, however, show non-trivial pattern: a U-shape behavior is seen with respect to change in mean burst size or FPT threshold whereas a monotonous trend is observed for change in transcription rate. Lastly, we also discuss how these predictions can possibly be tested via experiments on the lysis time of the bacterial virus bacteriophage λ.