22 hours in Munich

The S Bahn leaves every 10 minutes from the airport and takes about 45 minutes to get to the Hauptbahnhof.
There are all kinds of hotels close to the train station, including a Courtyard by Marriott City Centre (about a 5 minute walk.

A late dinner at the Augustinerkeller might be an idea. Breakfast at the hotel.

You would need to be back at the airport 3 hours before your flight, so plan on catching the Sbahn by 12:30 pm. That just leaves the morning for any sightseeing.

Totally following this. I was originally scheduled for less than 24 hours in Munich (arriving by train fro Kraków at 10:00am, departing by plane to the U.S. at 9:20am the next day). I found out that due to an unrelated schedule change on my ticket, I as eligible for a one-time, no-fee change to my itinerary. I used ot to extend my trip by 2 days and am so glad I did. Now I can breathe...

1) As someone said above, the S-bahn is easy to use.

However, be aware foreign (non-German) credit cards are not widely accepted. Be prepared with cash for shops and transit tickets. For instance, you can buy an S-bahn ticket at the airport with a foreign card, but not on-board trams or even at stations in the city centre. It's either cash or a German card.

2) 30-40 minutes from the airport to the city by S-bahn.

3) I got to the airport at 2pm for a 4pm flight with lots of time to spare! IIrc there was a separate queue for priority security but the wait time was 0 either way.

1. Easiest way to get between the airport and the city?

S-Bahn, as noted above. Either S1 or S8, whichever leaves first.

1. What time should we plan to head back to the airport for our 4:25pm flight? We are flying business class (thank you, airline points!), which I hope may give us some priority status for checking in and security?

The usual recommendation for transatlantic flights is to be at the airport 3 hours before take off. In most cases this is plenty, but at peak times it may well be necessary. There is of course a priority check-in for business class, but not at security.

1. Where should we stay, and what should we do, and (perhaps most importantly) where should we eat?

I would look for a hotel in the very center between Marienplatz and Rosenheimer Platz with easy access to the S8 (the area around the main station is not particularly pleasant).

Food - depends on your taste; there is everything and from all countries. If you want Bavarian: Hackerhaus (Sendlinger Straße) or Spatenhaus (Max-Joseph-Platz, opposite the Opera, a bit upscale); both are more pleasant and have better food than the better known beer halls like the Hofbräuhaus. A very good wine restaurant is in the Residenz (Pfälzer Weinstuben).

I got to the airport at 2pm for a 4pm flight with lots of time to spare! IIrc there was a separate queue for priority security but the wait time was 0 either way.

While the PP had no problems with a later arrival, the Munich airport authority recommends arriving 3 hours before an international, and especially intercontinental, flight. Just so you know.

S-bahn to Marienplatz as everyone else has recommended.

For dinner, I'd recommend Augustiner-Keller as someone did above as well. Right next to the train station. Great beer garden. Heard really great things about Wirtshaus in der Au but you'll need a reservation.

I'd get a hotel right at the airport. Then if you want to go into town take the S Bahn. Friday night in July, providing it doesn't rain, is a good time to sit outside and enjoy a meal in the old part of town. Find a beer garden, preferably with live music. The sun will be up until after 9pm.

In the morning you can either go into the terminal and drop your luggage or just have the hotel hold it until you get back.

Don't do the hotel breakfast, just take the train towards the Victualmarkt and get your food there. Pick up something to take on the plane too. Most of the tourist stuff is open between 9-10 am so you have a couple hours. If the weather is good you may want to consider walking the English Garden first thing.

I have used the Aloft and the Courtyard Marriott properties when traveling for business. They are close to the central Bahnhof and an easy walk into the old city.

The Munich MVV offers an "Airport-City-Day" ticket which is good for unlimited travel from
time of purchase until 6am the next day, including to/from the airport, for up to 5 people.

It's 30.50 EUR at the moment. I would buy that ticket in the airport transit center and then
you're good for the entire evening. If you stay in the city, you'd need another ticket to get
back out to the airport on Saturday. It is a pretty consistent ~45 minutes to get downtown.

The only hotel on the airport grounds is the Hilton. The Novotel is one subway stop away.
Everything else is in neighboring towns (Hallbergmoos, Erding, Freising) and will require
a shuttle or access to a S-Bahn stop. But since your main sightseeing time is Saturday
morning, staying downtown or something close to it seems the better option. Very nice
but not 5* hotels I could offer are the Platzl and Cortiina.

You have a few hours for sightseeing; I'm not sure I'd recommend any galleries or museums.
Stroll around the Viktualienmarkt, Marienplatz, maybe tour the Residenz. Dallmayr is a lovely
food boutique/cafe. Unless you're into touristy places, I'd avoid the Hofbrauhaus. Maybe just
walk in and take a picture so you can say you've been there. As noted, there are lots of other
Bavarian restaurant/beer hall places.

Since you are flying business class, you (assuming you're on UA or LH) will have access to
a dedicated security line. And if the Egates are working, you should be able to clear immigration
very quickly. So you may not need to get to the airport 3 hours ahead. But since you're in
business class, at least any extra time can be spent in the lounge. I will say that 4:25 in the
afternoon should not be all that busy for transatlantic flights, but it will be busy overall in July.

be aware foreign (non-German) credit cards are not widely accepted.

I have no idea where this comes from. We never had any issues with our US credit cards in Munich, or indeed anywhere in Europe. I used both my credit card and Google Wallet multiple times to pay for trips on Munich transit.

Agree with jphbucks. While some small shops and eateries may be cash only, I've never had a problem using Visa or Mastercard where cards are accepted, including ticket machines. Acceptance of Discover or Amex is much less likely.

where should we eat?

My favorite restaurant in Munich is https://paulaner-nockherberg.com/en/. This indoor restaurant is fabulous. They also have a beer garden in the summer.