Diese Garten ist für Publikum geöffnet.

English Translation

This garden is open to the public.

Is it not, “Dieser Garten”?

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Yes, you’re right.

I checked the original sentence on tatoeba, and I would suggest the following German sentence as an alternative from there:
“Dieser Garten ist öffentlich zugänglich.”

Sounds much more natural. Especially since public isn’t Publikum in German.
the public = die Öffentlichkeit
das Publikum = the audience

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And shouldn’t it be “fürs Publikum” and not “für Publikum”?

Yes, I wrote that first in my reply, but edited it out after.

Though all of them would be possible “für Publikum”, “fürs Publikum”, “für das Publikum” I usually would rather expect the latter ones.
It depends if it’s a specific Publikum or Publikum in general, but usually there is a specific Publikum which is why I’d expect “fürs” or “für das”.