How much time does a university typically give a professor to draft a syllabus and how long before they are approved?
Academic freedom means instructors can choose the material they teach.  Professor are loosely bound by the course calendar - if you are assigned a course on Greek drama and you teach instead Latin drama you may be reassigned - but otherwise you’re largely on your own: you can choose the Greek dramas to be studied in your course.

If the material is repeatedly too hard (high fail rate) then the Dean (usually in consultation with the chair) may reassign you but in a functioning unit some collegiality will usually resolve such issues, and will efficiently link courses in a pre-requisite chain.

There is usually an administrative procedure to make substantial changes to a calendar entry for a course.  Where I work it takes 2-3 months, mostly because the appropriate meetings (which must be done in a specific sequence) are rarely held “on the spot”.  The change is approved by the department, then by the Faculty, then scrutinized by some academic committee for justifications, additional resources etc, and if approved then onwards to Senate for final approval.

Completely new courses are usually tested out as “special topic” for,one or two years  to see what works and what doesn’t work.