The idea of Rome discovering gunpowder has been touted before, largely on YouTube to be honest. It's the sort of thing that would revolutionize warfare no matter what, and has been used in numerous atls to save rome. But what if the Eastern Roman Empire discovered the secrets of gunpowder under Justin the first, predessecor to Justinian? Would the byzantines weaponize it with canons, and fuel their reconquests of the west, or would it be a historical trivia bit not often thought of?
I had it be under Justin's reign since that would give Justinian his entire reign to use it
Inventing gunpowder and weaponizing it are two different processes. The latter would be a long-term, multi-generational affair with gradual innovation increasing the utility of gunpowder in warfare, but initially it isn't likely to be as decisive as the examples of 18th or 19th century armies facing enemies without firearms. Designing and building firearms on a large scale that don't explode in the hands of the wielder and reliably send even inaccurate shot toward the enemy isn't easy, and any early attempts at such weapons are likely to be inferior in most respects to bows and crossbows.
The actual time of discovery of gunpowder isn't as important as the time of its first military applications, which we can put in Justin's reign as per your post. This first use is probably going to be as fireworks for battlefield signals, or as one of several types of incendiaries in naval or siege warfare. The goal of such uses isn't to propel shot or blast the enemy with shrapnel, but simply to help spread fire, frighten horses, etc. Getting the gunpowder mix right to maximize explosive as opposed to incendiary potential would take time and experience, but it's not impossible for very simple bombs to be introduced as a natural evolution of incendiary use. Mixing some explosive pots filled with shrapnel in with whatever incendiaries you're flinging into the enemy fleet/city/siege equipment has some value, after all.
After all that, maybe you could get something like fire arrows and fire lances in Justinian's reign, but they're still going to be expensive specialty weapons. Part of the problem is that mass production of gunpowder and weapons that use them is expensive and capital intensive. The Eastern Roman state is centralized and powerful enough to feasibly begin producing it if deemed a priority, but scaling it up would take quite a bit of time. The good news for the Romans though is that this issue of supply means that many of its enemies will be unable to replicate gunpowder weapons in any meaningful number, with the possible exception of the Persians. Such a problem though is only likely to arise after Justinian's reign.
Ultimately, there's just not enough time for gunpowder and gunpowder weapons to develop if you want Justinian to have access to cannons and mortars, but it's not unreasonable for the army of his day to be using gunpowder fairly extensively in other forms.
