Hi Stacy -
I had a little more time to look at schedules in the SBB timetable and the lake boat schedule. In my reading, I do not find those boats referred to as ferries. The distinction may be important or have to do with exact translations, but only the boat from Beckenried to near Gersau, which carries automobiles, is called a ferry. In German, die Fähre or eine Fähre. By the way, the ferry landing station in Beckenried is some distance from the lake boat landing, about a km.
You can put together a route at many times during the day from Alpnachstad to Weggis by taking a bus with changes to Beckenried, and a boat to Weggis. You can also take a boat from Beckenried to Brunnen, and catch a bus or another boat to Weggis many times during the day. Look at the map and you'll see what I mean.
There is bus service from Alpnachstad to Beckenried taking 45 minutes every 30 minutes during the day in June. I picked the 19th as a random date. The route involves changes at Hergiswil and Stans with easy (6 or 7 minute) connections. The time table says that this service "Does not operate daily."
You could check the timetable for your preferred day; my guess is that service is limited on the weekends. Actually, I checked some weekend dates and there is service. So, I don't understand the comment, but you can check on the day you prefer.
Once at Beckenried, there are boats to Weggis approximately once per hour usually at xx32 from 1132 to 2032, plus one at 1000.
There are also approximately hourly boats from Beckenried to Brunnen mostly at xx27 from 0927 until 1627 plus one at 1800. From there, there are a few routes to Weggis.
Using the SBB timetable, from Brunnen, there is bus service along the lake shore to Weggis at xx27 and xx56, taking 40 minutes. There is some boat service at xx49 taking 1 hr and 26 minutes. There is also a possibility of train to Küssnacht am Rigi and the same bus in the reverse direction ( toward Brunnen) which goes through Weggis.
Those buses stop at Gersau Schiffstation, for "die Fähre", about 12 minutes after leaving Brunnen in the direction for Weggis., for example.
As noted, that trip via Beckenried will not be fast.
If you wish to make it even slower, with a nice funicular scenic ride, you can take a bus from Beckenried up to Seelisberg, and ride the funicular down to Treib, which is a boat stop for all the boats from Beckenried to Brunnen. Or, you can put a break into the route on the boat from Beckenried to Brunnen by getting off at Treib ( small tavern) and riding the funicular up to Seelisberg and then back down again. It is a nice trip.
I have stayed at Seelisberg a few times. Although I did not stay at this particualr hotel, the views across the lake from the terrace of the Hotel Sonnenberg ( up the hill a km or ride the bus two stops) are spectacular. See attached panorama. Also, at the left edge of the picture of Selisberg from a lake boat, you can just see the Hotel Sonnenberg.
So, if you wanted to take a couple of hours in additional in the afternoon to make that detour, you'd probably enjoy it.
A few photos are attached, including some from a Photobook that I prepared in 2009 about a trip with my daughter and granddaughter. The panorama with text was taken from the terrace of the Hotel Sonnenberg
On our trip, we also visited Riemenstalden, above Sisikon. If you go up there, take the Postbus up the hill. It is a steep narrow one lane road with turn outs for passing. If you are going up, you have to give way to those coming down, and back down to the last turn out to let them pass. ;-( The Postbus always has right of way.