Yes, of course. Many of the prophecies in the Old Testament certainly were fulfilled after the prophecies were written down. However, there are also Old Testament prophecies–especially in Jeremiah and Deuteronomy for which it is harder to prove that they were written down before the fulfillment.
Let me start with the ones that certainly were made prior to fulfillment. Because of the Septuagint translation of the Old Testament into Greek around 200 BC and because of the Dead Sea Scrolls from about 250 BC to AD 50, and also because of the Samaritan Pentateuch (500 BC) and also because of Jewish quotes from the Old Testament, we can assume without the slightest possible debate that the entire Old Testament was written at least two hundred years before the death of Christ. Therefore, any messianic prophecy in the Old Testament definitely predates the fulfillment. Therefore, Isaiah 7:14-15, Isaiah 9:1-7, Isaiah 53:1-12 and a rather long list of messianic prophecies in Isaiah certainly predate their fulfillment. Also, Jeremiah 31:31-36 about the New Covenant predates the fulfillment, as well as the prophecies in Jeremiah concerning the Messiah being the “branch” of Jesse (Jeremiah 23:1-8). Many dozens of messianic prophecies in Micah (Micah 5:2), Psalms (Psalm 22:16-18), Zechariah (Zech 9:9) and Daniel (Dan 9:24-25) also certainly predate the fulfillment.
But there are prophecies if the Old Testament which were fulfilled during the Old Testament. For example Deuteronomy 28-29 has some detailed prophecies of the fall of Judah in 586 BC. Jeremiah has the prophecy that the captivity would last for 70 years in Jeremiah 29 and elsewhere. Isaiah 42 has a prophecy that Cyrus would be the one to save Judah from Babylon. Ezekiel prophesies the destruction of Jerusalem in 586 BC. Joshua prophesies the ones who rebuilt Jericho would suffer death. It is nearly impossible to “prove” that these passages predate the fulfillment, as we do not have manuscripts which are that old. I believe that one of the reasons Ezekiel was taken as inspired is that he so precisely predicted the fall of Jerusalem. However, with Old Testament prophecies of Old Testament events, it will remain difficult to prove that these were made prior to the events they prophesied. Do we know for sure that Deuteronomy was written prior to the destruction of Jerusalem? Most scholars believe that the book Josiah discovered in the temple was Deuteronomy, and therefore that Deuteronomy predates the fulfillment of Ch 28-29, but there is at least an element of doubt.
In summary, there are literally hundreds of prophecies in the Old Testament that were fulfilled during New Testament times. In this case, we have solid proof that these prophecies were made prior to the events they prophesied. Even the hardened skeptic will have to admit that the Scriptures predate the events. Their tactic will be to deny that these were prophecies or that they were fulfilled by Jesus, etc. Well, they are wrong, but… With Old Testament prophecies fulfilled during the Old Testament (or between the Testaments, such as in Daniel 2,8,11), I believe that in every case the prophecy precedes the event, but I will not claim to have solid proof.