Regarding Zugspitze, the following web page shows the opening hours.
https://zugspitze.de/en/Service-information/Opening-hours-timetables
In the period May 9 to May 12, trips to the Zugspitze are possible. Please check the site shortly before you plan to go, as changes are possible. This only makes sense in good weather.
The closest mountain alternatives from Munich are:
Train to Lenggries and short bus ride to the Brauneck-Bergbahn (1:20 hours from Munich HBF). At the top there are moderate hikes, beautiful views and cozy alpine huts.
Train to Tegernsee 1:06 hours from Munich HBF. There are many possible hikes here, or simply enjoy the lake.
https://www.outdooractive.com/en/hikes/tegernsee/hikes-at-lake-tegernsee/1453397/
Don't miss the Bräustüberl Tegernsee.