Hotel in Munich

Hotel Gastoff Maisberger in Neufahrn. It is located not far from the airport, has ample free parking, the S1 station is out the front door and the trip into Munich is not too long. Prices are quite reasonable, rooms big, and an excellent restaurant.

How convenient is it to take the train to the city center and back? I haven’t been to Germany in years and a little nervous about getting around.

Why not stay at the hotel (Hilton or Novotel) at the Munich Airport and then pick up your rental car the next day (from the airport rental car office at the airport)?
The S-Bahn goes from the airport to the city center in Munich and is very convenient and easy to use,

The S1 line is very convenient and easy to use. The hotel I suggested is next to the station. You could also stay at the airport but I think the Hotel Gastoff Maisberger is a gem.

Hotel Platzl- we stay there every year. It’s a 1min walk to Marienplatz and three of the best markets

I'm starting to lean towards just biting the bullet and driving the rental car from the airport for 2 hours to Tirol. Not having been to a Christmas market before though am I missing out by skipping the Munich Christmas market and just seeing the ones in Innsbruck, Hall, Salzburg etc? Are they all similar to the one in Munich?

Here is my suggestion if you decide on that plan. Spend the night near the airport and get a night’s sleep before trying to drive after a transatlantic flight. Jet lag will make you impaired. It doesn’t mean that couldn’t do the drive. We know people frequently drive impaired. It does mean that the safest choice for you and the other drivers is to reduce that risk factor.

I'm starting to lean towards just biting the bullet and driving the
rental car from the airport for 2 hours to Tirol.

Reading between the lines here, it sounds like you're arriving from a transcontinental flight & are considering skipping the first fun place you will be in, namely Munich, where you could settle into the hotel suggested above & walk around a few of the Christmas markets. (Which is what I would suggest. Jet lag is not fun, I have to deal with it a few times when I've done that sort of driving straight off a flight. But it's always been somewhere I knew from previous visits, and never in the dark.) IF you arrive in the late afternoon you will be driving in the dark & it could be snowing, so will assume you're an experienced winter driver. The next day, you could take the train to Innsbruck, which looks like it's your second destination? OR stay 2 nights in Munich if you have the time to acclimate, lots to see & do! From Innsbruck if you need a car to visit smaller places, you could rent it there, and not have the hassle of driving back to Munich.

Having travelled a bit in the corridors you intend, if I was planning your plan I'd consider A) time of flight arrival, B) generally how okay or wiped out am I upon landing in Europe? and C) how much time do I have on the trip and where do I really want to be?

If an early arrival and if I generally feel okay, and if I really want to be in Tirol, drive to Tirol.

If a later arrival, and if I get hit hard by jet lag and if there is time stay a night in Munich.

Etc.

To the general Christmas market question, my instinct is that you'll be satisfied even if you skip Munich. But no direct experience there so grain of salt.

As for a hotel location if you pick up a car at MUC and stay a night in Munich, I'd offer that it will likely be easier to stay outside the inner city. The area around the Ost Bahnhof is just outside the part of the center you are interested in (pleasant 15 minute walk if you are a good walker, otherwise an easy, frequent and well-organized subway ride). It is close to the highway from the Airport, is a more car-centric area, and has many larger more car-centric hotels, many with underground parking. Seems easier than driving all the way into the core. Hilton Munich, Holiday Inn City Center Munich, Novotel Munich for instance.

Personally, I wouldn't stay farther out and use public transit to get in and out of the city for a visit, not for a short one day visit, particularly not on a first day. I like feeling like I've fully arrived on the trip, not like I'm doing a little 16 hour break sort of thing and my transit is not yet actually complete. For this reason, if I reasonably could just get the car, drive two or three hours and be in the place where it feels the most like my trip has actually started that's what I'd do. For me it would likely be a first night spent in Tirol.

Of course you need to be safe to drive, a call you need to make knowing yourself.

Have a great trip!

Will add that as Sandancisco notes above, rail connections to Innsbruck from Munich/MUC to Innsbruck are frequent, easy and fast, and Innsbruck is a big enough city that you could rent a car there as well. Easy Peasy.

If so you'll likely want to look at also returning the car in Austria to avoid usually significant fees to drop in a different country (if you are flying home from MUC). You could drop the car in Salzburg no problem and then easy train back to Munich. It's an options I'd certainly weigh and might choose.

@Carol now retired - thanks for that hotel recommendation. I am always on the look out for a fine place not costing the earth but logistically convenient.

I don't know the OP's market tolerance, but I went to two in Strasbourg a few years ago and that was more than enough. There are only so many ornaments you can look at and junk food and mulled wine you can drink. There aren't that many ornaments in the world, even though in old town Salzburg there is a smallish store that is chock full of them.