Usually when applying for time on a large telescope there are a few different categories of time. Most ordinary observations don't have to be done at any particular time and many of the larger observatories use queue observing for these targets. There's just a big list of these targets and the observer goes down the list and observes them one by one.

You can also apply for a "target of opportunity" when you don't know exactly when or where the event will take place, but there is a good chance that some event will occur over the next observing cycle. If you execute on a target of opportunity you jump to the front of the queue. If there's a particularly high time sensitivity (like for an FRB) then you might even interrupt an ongoing observation.

Things are trickier at a smaller observatory because they generally don't use queue observing. What happens instead is that an astronomer is granted some number of nights and they have the whole telescope for that amount of time. All you can do then is call them up and ask nicely. Usually they will though because they get on a paper without too much work. (The most highly cited paper I'm on happened that way!)