Travel from Munich International Airport to Neuschwanstein

Sounds exhausting. You could get trains from the airport to Fuessen in 3 hours, 3 hours return to the airport, maybe gives you 4 hours in Fuessen, certainly enough time to see the castle.
Buy 2 4-person Bayern tickets for a total of 86 EUR and that will cover 8 of you to Fuessen and back. You might store your luggage in lockers, return to the airport, retrieve your bags, then go to your hotel. You could be in bed by 10 pm, a long day.

I agree with sam. It really sounds tiring. It is possible and is a very long day after getting very little sleep on a plane.

We are arriving in Munich International Airport at 8:50am

No, you are scheduled to arrive in Munich Airport a 8:50. Assuming you are flying over the Atlantic, arrival time is never assured. So everything depends on ;your arriving on time.

I would never plan on renting a car and driving somewhere shortly after arriving. I've fallen asleep on the train right after arriving a few times. Thank God I wasn't driving.

Reserve a hotel for Freising. It's about a 20 minute bus ride from the airport. Go to your hotel, check in, go back to the Freising Bahnhof, and take the S-Bahn to Munich Hbf and the train from there to Füssen. If you arrive on time and go to Freising, then leave for Füssen at 12:10 (that gives you over 3 hours), you can arrive in Hohenschwangau at 14:55 (2:55 PM). Walk to the ticket kiosk and pick up tour tickets reserved in advance for after 4 PM. The castle is open for tours until 6 PM.

But someone in your party will be affected by jet lag. It will be a long day for them.

King Ludwig II's birthplace - Nymphenburg Palace - is right in Munich and far more interesting altogether than the late 19th-century palace he built far to the south. Something for everyone.

Carriage Museum
Porcelain Museum
Park/Gardens
Gondola ride
Eateries on premises and nearby

At Ludwig's Neuschwanstein, besides the hours of transport on trains and buses, you'll have just 30 minutes on the tour. And the walk uphill and back from the bus stop takes an hour.

King Ludwig II's birthplace - Nymphenburg Palace - is right in Munich and far more interesting altogether than the late 19th-century palace he built far to the south. Something for everyone.
At Ludwig's Neuschwanstein, besides the hours of transport on trains and buses, you'll have just 30 minutes on the tour. And the walk uphill and back from the bus stop takes an hour.

We were planning on visiting the Royal Castles of Neuschwanstein and Linderhof by way of a Day Tour from Munich and planning to pay nearly $250 for the tour for my family of 4. I looked at the link you shared and see the cost is only 11.50 euros regular for each person. I'm hoping my 18-year-old daughter and 16-year-old son will enjoy that just as much. My daughter is a photographer and may enjoy Nymphenburg Palace just as much possibly.

(This post attempts to address both the OP and jamescoste.)

From MUC airport, you can reach Nymphenburg in under one hour. Buy a group day ticket (€23.90) issued by the Munich transit authority (MVV) from a ticket machine at the airport train station. That covers up to 5 adults.

There are different routes for making this trip depending on what time you leave MUC. Use this page to find routes and schedules. You might get itineraries that use the S-bahn (train) + a bus connection or a tram connection. As long as the price for one (EUR 11.20) appears, your trip is covered for 2-5 adults using the group day ticket. (Depending on the ages and the number of children, you may need two group day tickets.) Perhaps you plan to go straight from the airport to Nymphenburg; in that case you can copy and paste the station names below.

From stop: München, Airport Munich
To stop: München, Schloss Nymphenburg

(not clear where you plan to stay, James, or which "FROM" stop you need. You might find your group day ticket cheaper depending on your location.)

"...we will most likely try to get lodging near the airport."

The Novotel Munich Airport is a nice airport hotel. If you stay near the airport, you probably want to drop bags at your airport hotel first, right? There's a local bus that gets you to the Novotel from the terminal. I don't recall the # but I'm sure the Novotel can inform you; it runs fairly frequently and stops at the corner out front of the Novotel.

To transit from the Novotel into Munich, you can take that same bus back to the airport, then use the trains from there, or you can make the walk over to the München, Flughafen, Besucherpark train station (not a bad walk without bags) which is walkable from the Novotel (we did it in around 10-15 minutes I think.) Your fare and ticket from this station will be the same as from the MUC airport station.

The same ticket is good for your return trip from Nymphenburg.

We have not selected our lodging just yet. We may try to stay close to the train station (possibly Hotel Royal). I do plan to buy the group day ticket (€23.90) (my kids are 16 and 18) from a ticket machine at the airport train station. We are not arriving in Munich until nearly 7pm but at least that will get us to the train station/hotel.

We are planning either:

• Royal Castles of Neuschwanstein and Linderhof Day Tour ($250 total or do I try to plan this myself for cheaper?)
• Nymphenburg Palace
• Munich WWII Combo: Dachau Concentration Camp Memorial Site and Third Reich Walking Tour ($140 total). We really only want the Dachau Concentration Camp tour and are not as interested in the Third Reich Walking tour. Is this also something for which I can arrange transportation and pay the entrance fee more easily/less costly?

We are only in Munich 2 days (arriving late on 7/10) and leaving late on 7/12 to Salzburg.

We really only want the Dachau Concentration Camp tour and are not as
interested in the Third Reich Walking tour. Is this also something for
which I can arrange transportation and pay the entrance fee more
easily/less costly?

Dachau is so easily visited on your own from Munich downtown that it amazes me that so many people spend all the extra money for a 3rd party tour. Everyone who has paid the high price of a 3rd party tour claims that it was worth the extra cost, but I've never heard of anyone doing both and comparing. The memorial's own tour lasts 2½ hrs plus there is a half hour movie shown multiple times during the day, 5 times in English. What more can you learn in more than 3 hours?

The guides from the Memorial undergo extensive training by the Memorial. I'm sure they tell you everything the Memorial thinks is important.

To go to the Dachau Memorial, which is in the second zone, from downtown Munich, you should use a München XXL Day Pass (Tageskarte). You will probably want an Innenraum Tageskarte anyway; the XXL Tageskarte is only 2 -3 euro more for single or group respectively. This ticket will cover round-trip from downtown on the S2 plus the bus (#726) from Dachau Bahnhof to the Memorial. The bus stop at the Bahnhof is well marked. The actual tour by the Memorial's own guides cost 3,00€/adult. English tours are at 11 am and 1 pm.

I appreciate the info. I'll look more closely into planning my own trip to Dachau!