If I were writing the sentence, my first correction would be to change from Tuesday to on Tuesday. I find that the use of from to be a lot more unnatural than the plural posts.
Having said that, while the plural posts is possible, it doesn't sound as natural to me as the singular would.
However, even though there might be a plural number of posts, they will still all be for an associate professor position. So, the singular professor does need to be used. Turning both of them into a plural is a mistake.
In short, I would write it like this:
Entrance exams for the post of associate professor will begin on Tuesday.
While my preference is for the singular, I would be fine with the plural posts if there really were several posts available.
If there are several posts available, you can look at the sentence in the following way to understand why the singular professor should be used:
Entrance exams for associate-professor posts will begin on Tuesday.