With all managed packages, even if you don't have access to the trigger, you can still write your own trigger for the specified object.
You can control "when" the record is updated by making sure to specify which trigger (before update, before insert, etc...). That being said, there's no way to control if your trigger is fired first of if their trigger is fired first so you will want to be conscious of this in the event that it doesn't work exactly as you envisioned it.
I would also advise familiarizing yourself with the order of execution of triggers. You might be able to achieve your goal via a PB or Workflow which could also change the order in which the tasks execute (notice in the article how it mentions that updates via wf or PB fire the trigger a second time).