That's possible, and very easy.
so is it basically just like taking the train within Germany then? Do I even need to bring my passport?
Sadly, it is not just like taking a train within Germany. In theory, it should be. In practice, due to developments since 2015 onwards, no. You may be, and should expect to be controlled at the border. And you had better had a passport on you then.
Regardless, you should carry your passport anyway. While at least in Germany there is no general obligation to carry ID at all times (or passport in your case, since you are not a Union member, so you have "Passpflicht", not "Ausweispflicht" -- which means ID is not sufficient), you are required to be able to prove proper ownership of a passport and you are to present it upon being asked by police in due time. Well, whatever that means... in due time (it's not defined, I am assuming something like "Sure, let me go inside my room and fetch it" would do, but "Uh, I left it in another city" might not). Not having your passport on you doesn't mean you're going to get sentenced to death, but well, it means trouble which is not really necessary during your holiday.
Plus, when crossing borders, carrying your passport is a strict requirement.
somewhere in Austria (Innsbruck maybe?) so I could see the Alps.
If the intent is to literally see the Alps, you can do that just fine even without leaving Munich. Or, you can do it from a ton of very lovely nearby places all over the region which are easier to reach (Tegernsee or Chiemgau come to mind, or even some nearby places which you can actually reach by subway or suburban train, Munich area is quite awesome in that respect).
I'm not saying you shouldn't go to Innsbruck, it's definitively a good choice. Though depending on your luck, and depending on how late you buy, you may be surprised how much Deutsche Bahn can charge for a ticket.
Also, there's actual Alps in Germany (think e.g. Zugspitze), so you do not absolutely need to cross a border to see (in a sense of walking them) the Alps. It's somewhat closer (half an hour train time less), too.
If Kufstein (Austria) would also do instead of Innsbruck, you can possibly save quite a bit of money by getting a "Bayernticket". That's a very interesting option in any case because it's a 24-hour do-what-you-want ticket (no ICE, all within all of Bavaria, plus almost -- but not quite -- to the Lake Constance (2km short, that's probably on purpose!), plus some locations in Austria -- Kufstein and Salzburg among them, but not Innsbruck) for a very reasonable price, and if you travel with family or friends, you can add them to the ticket for a small extra charge, which makes it very worthwhile.