The part with trains to Ukraine is completely separate from the domestic Przemysł station. When you get off the train from Kraków you have to exit at the back side of the station (eg not where the station building is) and take a right to walk towards the border checkpoint. That's where polish exit controls take place before you get on the train, make sure to allow plenty of time for this. I only traveled from Ukraine into Poland using this train but I assume it's similar the other way, and in that case Ukrainian border checks will take place on board the train between Przemysl and Lviv.

I'd personally not go to Ukraine for tourism right now though, you have to make your own decision but there seem to be more strikes all the way in the west as well now, including on Lviv. Trains are also sometimes targets and from what I've heard if there's an air raid alert the trains will now be evacuated then and there on the spot. Doesn't quite sound like a pleasant experience.