A Bayern Ticket is a one-day travel pass that covers local trains (not
ICE, IC, or EC long-distance trains)
You really have to go out of your way to take a long-distance train to get to Füssen. The direct route uses one or two regional trains.
The really sneaky thing is that DB schedules the first after-9 train to Füssen to leave Munich Hbf at 9:52 AM (almost 10 AM), and it gets to Füssen at 11:55. However, the 8:53 regional connection to Füssen (with a 10 minute, same platform change of trains at Kaufbeuren) makes one stop at Pasing, still in the inner zone of the MVV at 8:59 and leaves Pasing at 9:00, so the Bayern-Ticket is valid from Pasing. A single trip, inner zone MVV ticket is 2,90€, for for an additional 2,90€ per person, you can leave for Füssen at 8:53 and get there at 10:55. The connecting bus then gets you to Hohenschwangau at 11:13. It's less than a 5 minute walk from the bus stop to the ticket kiosk, so you can schedule your Neuschwanstein tour for 13:00 and pick up your ticket 90 minutes earlier at 11:30.
For a tight schedule like Munich to Neuschwanstein, I would prefer the train option. I have found it to be very reliable. If doing it by car, you need to provide a time buffer. If you experience a traffic tie-up or a wrong turn, you could miss your ticket pick-up time.
We are also planning to fly into Munich and staying the night in
Fussen and going the next morning to the castles.
Then using the Bayern-Ticket before 9 AM is probably not an issue. You can buy the Bayern-Ticket at the airport and use it for the S-Bahn into Munich as well as the train to Füssen. Depending on where your accommodations are in Füssen, you might need to use a bus in Füssen, and that should also be included in the Bayern-Ticket. The next morning, a bus ticket to Hohenschwangau will undoubtedly be cheaper than any pass, but if you are returning to Munich that night, a Bayern-Ticket should be purchasable on the bus (exact change appreciated) and would cover round trip to the castle and return to Munich by train and any use of conveyances in Munich.