Does Islam support this statement? Are there any verses in Quran or Hadith regarding this?
The statement you re referring to (in the YouTube video) is "Either a woman gets married, or become public property."
In Al-Mughni (Hanbali school of jurisprudence), Ibn Qudamah says:
وأما السنة فقول النبي صلى الله عليه وسلم: «يا معشر الشباب، من استطاع منكم الباءة فليتزوج، فإنه أغض للبصر، وأحصن للفرج، ومن لم يستطع فليصم، فإن الصوم له وجاء». متفق عليه. في آي وأخبار سوى ذلك كثيرة. وأجمع المسلمون على أن النكاح مشروع. واختلف أصحابنا في وجوبه ـ فالمشهور في المذهب أنه ليس بواجب، إلا أن يخاف أحد على نفسه الوقوع في محظور بتركه، فيلزمه إعفاف نفسه وهذا قول عامة الفقهاء
— NOTE: My own translation, so treat with care:
As for the Sunnah, the saying of the Prophet ﷺ: "O' young men, those among you who can support a wife should marry, for it restrains eyes (from casting evil glances) and preserves one from immorality; but he who cannot afford It should observe fast for it is a means of controlling the sexual desire," agreed upon, as well as in many other verses and narrations. Muslim agree on the permissibility of marriage but differ in its mandate. What is commonly agreed upon in the madhhab is that it [marriage] not a mandate unless one fears of committing what is forbidden then one must do what is required to keep oneself chaste, and this is the opinion of most of the jurisprudents.
This is the same view adopted by other schools of jurisprudence, and the majority of jurisprudents. Marriage is not a mandate, neither for men nor for women, but is strongly encouraged. What this means is one should definitely seek to marry, but one will not have sinned if one does not marry.
The opinion of "becmnig public property" is that of Dr. Zakir Naik, and not one that is supported by Qur'an or hadith. The reason behind Islam allowing up to four wives is undeclared to us. Any opinion on the matter is that of the person making the statement. In most cases, I find such opinions to be fallacious (e.g., the false dilemma — either/or — either a woman gets married, or she becomes public property), but this is strictly my own personal view, and is not meant in any way or form to be that of the person; rather, their opinion.
Note that the hadith quoted above is in Sahih Muslim, Book 16, Hadith 3, as well as other books of hadiths' collection.
What is the upper age limit for a woman to get married? Because a woman unmarried now may be married tomorrow, so when she could be public property?
There is no declared upper age limit for a woman (or a man) to get married. It is a matter of ability to support a marriage.