I initially had some trouble booking this too far in advance. The site kept telling me that there…read morewere no tickets available, but it wasn't clear if I was trying to order them too early (several months out), etc. Eventually I must have entered the active window as I was able to see dates / times; this was probably 2 - 3 months before our trip.
Your options for seeing the Lipizzaner horses are, IIRC, a live show, an exercise / rehearsal session, or a behind-the-scenes tour. Prices go down and flexibility goes up as you progress through them in that order. Giving our schedule, I opted for the tour. You also have to pick your language at booking time, which makes sense. You are given a specific time for the tour.
The venue is in the Hofburg complex, in the same 'tunnel' area where you can buy tickets for the Sisi Museum. The entrance for the Lipizzaner events is across the road. It is not that obvious, but they had a TV running video of the horses, so that helps. There is a desk where you check in and then you wait in a lobby area for your guide while the rest of your tour party assembles there. They have informational displays about the horses in this area.
Our tour started with seats at the main performance arena. The guide went over the history of the place, how things are run today (i.e., how you become a rider, etc.), where the horses come from, etc. It was interesting. The arena is fairly large.
We then progressed to check out an adjacent training pavilion just outside. After that, we crossed under part of the building and across a road to a courtyard where you could actually see horses peering out of some stalls. The guide would stop occasionally to continue his narration and to answer questions. We then entered a large room where they stored bridles and saddles for the horses. You were free to take photos up to this point. However, you cannot take photos when in the enclosed area with the horses.
We spent a few minutes in close proximity to the horses. The guide discussed things like how they varied in size and how their color evolved as they aged, plus how their naming represents their lineage, etc.
We then emerged back into the courtyard where the tour concluded. Here, you are just a couple minutes' walk from where you started. There is also a small gift show nearby if you are interested.
Everyone in our party enjoyed the tour and I was glad we booked it. Recommended.