Sarah got to the station just on time to catch her train to the airport.
Why do we use "her"? Does it mean Sarah already have the ticket in advance? if not, can we omit it?
In English, "possessive" forms (including the Saxon genitive 's as well as possessive pronouns) don't necessarily imply anything about "ownership".
OP's example is a perfectly natural usage that simply means the train associated with her in some way (usually, just the specific train she intended to catch, whether booked and paid for or not).
Although it's not central to the question, I'd also like to address the point commented on by @Teacher KSHuang above (we usually say just "in" time, not just "on" time). I think this chart is interesting...
As you can see, there's nothing particularly unusual about [he arrived] on time (there's one of those for every two instances of [he arrived] in time), but if we include just, we find the in version is over a hundred times more common than on.
That huge difference in prevalence means most native speakers will at least notice OP's relatively uncommon version, but it's not easy to explain why we don't normally use it.
I think it's because to be on time usually implies to arrive at the time you are expected (to be punctual, neither early nor late), whereas to be in time usually implies to [narrowly] avoid being late.
Because of that, I think in to arrive just on time the adverb might tend to be interpreted as meaning exactly (arrive at precisely the expected time, be very punctual). Since that clashes with the in version, where just = almost not, barely, we tend to avoid it.