You have a few options. There are two SEN/*G lounges on the schengen level with standard SEN lounge ammenities: showers/beer/good food/internet terminals, one opposite G24 (larger) and another opposite G28 (smaller), quite close to each-other.
On the non-schengen level there's a lounge opposite H24. I've never been to this one so I can't comment, but I'm sure it's like the other ones, since all of LH lounges are pretty much the same.

When you land, you'll be on the Schengen level, so you'll have an option to visit all 3 lounges .

There's a beer garden between T1 and T2, but I wouldn't go out, as LH has MUC beer inside the lounges on tap.

The train into town takes 45 mins each way, so that's out of the question for you.

Here is LH's description of First Class services at MUC