Is there a prophecy of when and where Jesus was born?
Yes, definitely there is. First of all, Micah 5:1-5 is a prophecy that the Messiah will be from Bethlehem Ephthrah. "But you, Bethlehem, though you are small among the clans of Judah, out of you will come for me one who will be ruler over Israel, whose origins are from old, from ancient times." There were two cities named Bethlehem in Israel. Jesus was born in Bethlehem Ephthrah. This passage describes the one to be born in Bethlehem as the Shepherd of Israel and the Peace of Israel. This is a clear messianic passage.
Second, there is a prophecy of when the Messiah would come to Jerusalem. This one is a little bit harder to get at first read, but it is an even more amazing evidence for Jesus being the Messiah. It is Daniel 9:23-26. This is a prophecy of the coming of the Messiah to Jerusalem. It describes one who would "finish transgression, put an end to sin, atone for wickedness, to bring in everlasting righteousness, seal up vision and prophecy and anoint the most holy." Of course, the Hebrew word for anoint is Messiah. When will the Messiah come to Jerusalem? This passage says that it will come "seventy sevens" after the decree to restore and rebuild Jerusalem. Seventy sevens is 490 and we can be confident (given what is below) that this is 490 years. How can we be confident of this? Consider the "decree to restore and rebuild Jerusalem." It is found in Ezra 7:11 and following. This decree was given in the seventeenth year of Artaxerxes, which is 458 BC. Four hundred and ninety years later is AD 33 (noting that there was no year zero AD). So, Daniel predicts that the Messiah will come to Jerusalem about AD 33. Actually, if you read the passage in Daniel carefully, it has the Messiah coming to Jerusalem during the last "week," which means sometime between AD 26-33. In fact, Jesus was probably crucified in AD 30 or 29. This is a truly amazing prophecy of the Messiah which Jesus fulfilled in a rather spectacular way.
So the answer is yes, there are prophecies in the Old Testament of both the time and place of the birth of the Messiah. If you are interested in more on messianic prophecies, there are two chapters on the subject in my book "From Shadow to Reality" available at www. evidenceforchristianity.com.