24 hours in Tegernsee

I stayed in Tegernsee back 8 years ago or so, not sure if any information I have is dated. We stayed right in Tegernsee, train station not a bad walk to the town center, for a short stay that would probably make the most sense. To be honest, I don't recall the name of the hotel.

We also took the cable car up, and hiked back down. Nice walk, not too strenuous, there is a restaurant at the top, then along the way down, there were two places for drink and a snack.

At least some restaurants will be open on a Monday. There is a very good brewery/restaurant, Braustuberl Tegernsee, that is open daily.

There is a wonderful Sauna complex on the lake if you are not too shy (it is textile free, like most suanas), great after a day of hiking.

The train is the BOB, a small regional railway, the schedule is dependable, no need to buy tickets ahead, it is basically a regional commuter train, I doubt the schedule changes much, if any. Do be aware of when the last train leaves Munich, and at least some trains that leave Munich, split midway, with one half going to Tegernsee, the other half elsewhere. Just need to make sure you're on the right half.

I think you will be fine mid September. I was last at Tegernsee the first week of November 2023. That was the last week where we stayed was open until ski season. The place we stayed has Ruhertag Monday and Tuesday, so that won't help much (Cafe Restaurant Angermaier). Another choice was already closed for the summer season then (Gästehaus und Ferienwohnungen Gutfelder Hof). Bavarian school vacation is over 9 September, so I would think you are there at a good time. Yes Rottach-Egern is pricey and you might see a few Ferrari, Porsche and expensive Mercedes etc. while there. I think that for one-two overnights it is the place to stay. You might look at someplace farther from the lake towards the south/south east as we did last visit. Many holiday apartments there. I drive, but we did use the 360 bus a few times. The Tegernsee Card that comes with some stays is a good deal. Poking around I found this: https://efa.mvv-muenchen.de/index.html#trip@enquiry

take the cable car up Wahlberg mountain and maybe walk down,

The Wallberg offers a beautiful view of Lake Tegernsee, but is also quite frequented because of the cable car. If you want to take it quieter and enjoy the beautiful alpine landscape, then take a taxi to the Klamm/Königsalm car park (not: Siebenhütten!) south of Kreuth (1) and hike up the paved path to the Königsalm (ca. 1:30), a former hunting lodge of King Max of Bavaria (closed Tuesday). It is situated at an altitude of 1200 metres in a beautiful high valley directly below the Blauberge mountain range. They offer fresh milk, buttermilk, butter and cheese bread and a simple type of sheet cake. You can sit on tree trunks in the shade or in the sun and enjoy life. Back in the valley, you can take the way back along the Weissach to Kreuth, where you can catch the Tegernsee bus (the hotel will give you a TegernseeCard, which entitles you to free bus travel).

and at least some trains that leave Munich, split midway,

All BRB (formerly BOB) trains to the Oberland consist of three parts. The normal composition is: Bayrischzell at the front (splitting off in Holzkirchen), Lenggries in the middle (splitting off in Schaftlach), Tegernsee at the back. However, there are always staff on the platform (in the NW annex to the main hall, track 27-36) to manoeuvre misguided North Germans (and occasionally Americans ;)) into the right carriages.

but are there many taxis or buses otherwise?

There are two bus lines that circle the lake, one clockwise, the other counterclockwise. Both depart from the Tegersee station. You can find all bus lines in the region here; search the pdf for "Ringlinie A" or "Ringlinie B".

(1) There are also infrequent busses (#9556), the name of the stop is Wildbad Klamm, Kreuth

Thank you everyone, I don't know how to respond to each individual post but that's some great advice!