Yes, it's a very common expression and sounds totally natural in this context! "Short on" is a prepositional phrase that suggests a lack of a necessary resource. You can be "short on" lots of things--money, patience, space, energy, etc... It's definitely idiomatic, so it makes sense that it would be confusing.

It's a humbly nonchalant and common-but-imprecise way of saying 'lacking time'. Still, the words are so simple that every native speaker should understand. But in case someone doesn't understand, then 'short of time' would be a more precise way of saying it, and lacking time even more precise, but more fixated, less nonchalent, thus less humble.