Hi KiwiMarg -
<<"So if we were to cogtrain to Rigi Kulm and hike around there, using your suggestions, and then take train down to Arth Goldau, and then cable car back up to Rigi Scheidegg. This would save us the 2 hour walk from Rigi Kaltbad to Rigi Scheidegg. we then walk on to Urmiberg and Brunnen. Would that save us much time, and energy, do you think?">>
I'm not sure how long a day and how much trekking you want to manage.
Check out the timetable for individual sections of your plans. And, study the maps carefully.
The earliest boat from Luzern is at 0812. Arrive Vitznau 0909. good connection to the cograilway.
Train to Rigi Kulm 0915 till 0947.
Hike one hour to Rigi Kaltbad ( I'd take longer than that because i look around and take pictures at Känzeli and along the route. Less if you follow the main trail and don't divert at Rigi Staffelhohe.
Look at the map - there is a short cut to First from Staffelhohe via Oberer Firstweg. Saves some altitude change but I only did it once. I like the scenery from Känzeli. (By the way, these days, First is nothing but a trail junction and a few buildings.)
The map also lets you see how the old rail line ran from Kaltbad to Scheidegg; it is obvious from the trail contour.
Arrive Kaltbad 1047?
Lunch? 20 minutes if you bring a sandwich.
Leave RK and walk 5 + hours to Urmiberg Talstation arrive 1615.
If you want to use the transport down to AG from Kulm, the lowest point at which you can get it is Rigi Staffel, the next station below Kulm, where the rail lines diverge. The trip down to AG from Rigi Kulm takes 51 minutes. The station for the cable way to Rigi Scheidegg is at Rigi Kräbel, a bit above Arth -Goldau, and 32 minutes from Rigi Kulm. It may be "halt on demand" stop; not sure.
I've done that trip and I recall a need to move fast to catch the cableway...tight connection. It runs every 30 minutes at during the day at xx55 and xx25.
So, if you timed everything to catch the transport(s) you would save about an hour.
To be sure, I just checked the online timetable and also the arrival and departure panel and can find no service in August. Hmmmn. Looked it up in my paper copy and found that they are renewing the cableway from March 20th until sometime in the Fall. So, no go this year.
<<"And are you suggesting that the bus from Urmiberg Talstation is the way to go back to Lucerne, rather than a train. It does seem easier if we are at a bus stop anyway.">>
No. I am suggesting that you look at timetables and the map to see which route is fastest. Both involve a bus and a train. There is likely more service from Brunnen to Luzern than from Küssnacht am Rigi. Brunnen is a stop on all the trains along that route. Küssnacht am Rigi is a smaller and more "local" station. Just guessing on that one. I'm sure that going through Brunnen is faster if you make reasonable connections, because the bus in the other direction has too long a trip. Scenic, though, along the lake.
However, you can walk from the Talstation to Brunnen easily. The other direction is too far to walk.
Please look at this link.
map.search.ch/?pos=690564,206124&z=64 &poi=bergbahn,halteste lle,zug
The cableway at Urmiberg is a small time operation. It has 6 passenger gondola cars. (They claim 4 passenger capacity, but they can hold 6.) Runs twice an hour at xx05 and xx35 ( the sign says "approximately.) One car goes up and one car goes down each half an hour. If they have alot of people, they run extra trips.
All signage at the Talstation is in German. There was no live person there. The cable cars are controlled form the Bergstation. If you go upwards, you cannot pay until you get to the top,.
The guy at the top who runs the cableway has a video hookup and can see you in the lower station and make announcements...such as when the car is about to leave. A mysterious voice tells you in German that it is time to get in the car. Since one goes up while one goes down, the one going up may suddenly close its doors and go up; there is no one there to get you into the car. At the top, there is someone there to take your money and to get you into the car.
This particular part of your route is off the beaten track for most tourists. It does not have all the English language information that the bigger stations have.
I can't find the time of the last trip down in the timetable. I'm guessing about 1830.
( I just looked up their web site. It is at 1800.)
You should make that easily ,but if you are slow hikers, I'd worry.
Once you are down, you've got a 7 minute bus ride to Brunnen....3 times per hour at xx08, xx38, and xx53. Then, about an hour train ride back to Luzern. You can check those connections.