Of course the Koran was written well after the texts that were to make it into what became the (Christian) Bible in the 4th century CE.
In the begining of Islam, in the Arab Peninsula, there were many Jews as well as some Christians: the important events and characters of the Bible were common knowledge. Adam and Eve-Hawa, Noah-Noh, Abraham-Ibrahim, Moses-Musa, David-Dawud, Solomon-Suleiman, Jesus-Issa, Mary-Mariam, Joseph-Yusuf and others are present in both books, although the events depicted may differ.
Mentionning Noah specifically, one may be interested to know that he and the flood story were inspired by a much earlier story written by Sumerians: the Epic of Gilgamesh, the oldest known book.
The New Testament says Jesus is gonna send the Holy Spirit after his departure. That's how Christian understanding goes. The Greek word in John 16 5-7 is "Parakletos" which means "one who consoles or comforts, one who encourages or uplifts ". Islam and Manicheism have interpreted it as announcements of the forthcoming prophets Mani and Muhammad.
Not long ago, the discovery of a new/complete version of the Gospel of Barnabas made the buzz in Muslim countries as it confirms the islamic perception. This text, among many others, was not taken into account by the official church leaders who made the selection for the "offical Bible". But that version of Jesus life was popular in some parts and fitted in with Islam.
Early Jews, early Christians and early Muslims share some common cultural background, influenced by major civilisations: Sumerian, Phoenician, Babylonian, Egyptian, Persian, Greek, Byzantine,...
Their Holy Scriptures intemingle, recycle, reinterpret these heritages.