At the beginning of the school year, students are required to take measures that will include the standard University Center’s assessment battery.
Should it be Center's or Center?
Either can work. Both forms convey essentially the same information.
With the possessive, the phrase is understood in the straightforward manner: it's the assessment battery related to (owned by / administered by / etc) the University Center.
Without the possessive, it makes University Center an attributive noun.
In grammar, a noun adjunct or attributive noun or noun (pre)modifier is an optional noun that modifies another noun; it is a noun functioning as a pre-modifier in a noun phrase. For example, in the phrase "chicken soup" the noun adjunct "chicken" modifies the noun "soup". - wikipedia