Hi heytekc -
Annika has suggested one possible alteration in your journey. It would result in your using the same boat route twice, but, other wise is a very scenic and more relaxed option.
If you would like to spend a little tiome to consider alternatives, there are enough travel options around the lake that you could figure out many possible combinations.
It is somewhat like doing a jigsaw puzzle, but you can fit the pieces together in many different ways.
Use timetable, plus this map:
map.search.ch/?x=23832m&y=16656m&z=8& poi=bergbahn,haltestel le,zug
In the menus on the map I have turned on symbols for public transportation. If they are too confusing, select points of interest, then traffic and uncheck the boxes.
All parts of the lake are beautiful, but I prefer the southernmost part, the Urner See.
For some images of the area, please see this link:
I find it helpful to look at the boat schedules separately from the timetable, even though you can make the timetable show the boat journeys. For boat schedules on Lake Lucerne and some maps ( I recall) see this link:
Note that there are several different routes and combinations available.
One idea, out of many possibilities, is to travel from Luzern to Flüelen by train. Catch a lake boat to Vitznau and then finish your journey by going up to the Rigi from Vitznau and descending via cog rail to Rigi Kaltbad and cable car to the Weggis valley station, followed by a 10-15 minute walk downhill to the boat dock at Weggis, with a boat back to Luzern. That covers almost all of the lake boat route; you would only miss a very short stretch between Vitznau and Weggis. I think you could put that together within your time constraints and see all of the sights you wish to see, except the trip up to the Rigi from Arth-Goldau. In my opinion. that is the least scenic of the three options. The critical timing would be to get the boat from Flüelen to Vitznau.
Her is another, slightly more complicated option.
Take the train to Flüelen, boat to Brunnnen. Shuttle bus or ten minute walk from lake boat dock at Brunnen to Brunnen train station. Short train ride to Arth-Goldau, then finish your journey as originally planned. I think that can fit within your time constraints, too. You miss the short stretch of the lake boat ride from Brunnen to Vitznau, That is abit more than yu miss on the first route that I suggested.
Both of those ideas depend on making good connections with no more than short delays. Check them with the timetable. I did not do so, but I estimate that they should work. The only way to be sure is to work through them with the timetable. They both save time because they eliminate taking the lake boat on the northern part twice in two directions; they only take it once.
In any case, Annika's advice about being back in Luzern by 1600 is a good target for any travel plan around the lake.