From the Walserberg border crossing to the Salzburg-Nord exit, no motorway toll sticker is required. You need a motorway vignette to use the motorway beyond that.
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You would be better off keeping it in Munich. You have to take the passport with you. That there are no (regular) border controls does not mean that you can cross borders without passport. If you go without passport you're entering Austria illegally.
An adhesive vignette can be purchased in Austria at ASFINAG distribution points, mainly at petrol stations. In Germany, adhesive vignettes are usually available at petrol stations on highways and petrol stations near the border, as well as at ADAC offices.
Driving without a vignette is quickly fined, usually €120. The price is €9.60 for a 10-day motorway stamp in Austria. Austrians are pretty strict and uncompromising when it comes to complying with laws and traffic regulations, so don't get caught out!
There are 10-day, 2-month, and annual vignettes. These stickers must be visibly stuck to the front windshield. You can purchase vignettes online on the ASFINAG website. Alternatively, you can buy a vignette from gas stations in Germany.