Here's your parameters: you have 4 hours in an afternoon to spend in Luzern, starting and ending at the train station. This can't be changed. It's late April. What do you do and see? Art and museums are not options. Go!
With such a short time I would probably just walk around the town, walk across the Kapellbrücke, go see the lion monument, and if the weather is nice maybe a 1-hr sightseeing cruise on the lake. Sit in a sidewalk cafe along the river and enjoy the ambiance, it's so beautiful there. You say no art or museums but how about churches? The Jesuit church is a really nice baroque church from the 17th c.
I'd suggest taking one of the walking tours which will cover the history and some of the significant sights including the "Lion of Lucerne" in some cases. Some tours leave from the station and they run about 2 to 2.5 hours, depending on which one you choose. For the remainder of the time, stop at one of the restaurants or Pubs along the river (close to the station) and either have lunch or just a pint.
There's lots to see in that area, but with four hours you won't have time for much.
And definitely pop into the Jesuit Church as it sits right on the River Reuss.
Just contain yourself in the downtown medieval city center strolling around and thinking "how does such a beautiful place like this even exist".
Agree with the others, walk out of the station to the boat docks in front of the station, see if there is a short (~1 hour) trip about to start. Walk over the river in to the old town, along the river and the Kapellbrücke (wooden bridge). Skip the lion, it is some way away and not that impressive. There is another smaller wooden bridge further down the river at the end of the old town, where the walls are.
If you fancy it you can walk along the top of the walls for a short way. It may be a peaceful town now, but it must have been a rough neighbourhood a few centuries ago for the town to devote so much resources into defences, the two wooden bridges were part of the defences, to stop invaders circumventing the walls by water.
Wander back into town, and have a break at a café, lots of them, the ones by the river are more expensive.
That should easily fill up 4 hours.
Thank you everyone, great suggestions all. Sounds like our short layover will be worth it!
To add or bring together many of the great suggestions...
Yes, cross the both old bridges (coming and going), check out the old town while coming back from the the wall on the ridge 'above' the old town and, while there, walk the length of the wall (and maybe back).
So cross the first old bridge you come too past the modern one and take it across to the old town. Take a right and go down the road/river until you turn left to head up Lowenstrasse. While heading up to the wall entrance stop and see the Lion, off to the right behind some tourist shops. Then continue up to the entrance of the wall and follow stairs up onto the wall walk way. Head down left to the end and then down the stairs and you are out (20-30 mins maybe). Head down the path, past the llamas, and take a left into the old town. Check it out and then, when you are ready head back across the river and to the train station, walk over the OTHER old bridge and take a left back to the train station.