That's absolutely correct. The fast ICE trains Cologne-Frankfurt-Nuremberg-Munich practically all pass through the long distance station at the airport.

I have always preferred the skyline train from T2 to T1 because of the apron views, but that's maybe not relevant for you Once you arrive at T1 I would say you need approx 10min to the long distance train station. On that note, I would recommend to always have a reservation for the ICE trains as especially the 1st class tends to be full.

Not sure if you ever used a German high speed train, they are actually quite comfortable but tend to have some quality issues like non-working coffee machines etc. I still find it a good experience compared to many other countries, but Germans love to complain about Deutsche Bahn (IMHO on a high level). Using LH to Munich is actually not much faster (given how far outside MUC is and given that the tube takes 50min to central Munich).