The letters to the Corinthians where finished in 55 C.E. So any of the scriptures that where finished or even partly written at that time could be what he is talking about.
Obviously the Old testament was finished, so really we need to look at the books of the new testament and when they where written in comparison with the finished date of Corinthians.
By 55 C.E. the books of 1&2 Thessalonians(50 & 51 C.E.) and the book of Matthew(41 C.E.) where completed. Mark(finished 60-65 C.E.) and Luke(finished 56-58 C.E.) could have started being written at the time, but I don't know that for sure. The testament of John and Revelation where not written until the end of Johns life. That was WAY after these letters where written by about 30 years. So I know Johns testament wasn't included when Paul was referencing the scriptures.
It is possible that Paul knew what would be contained in the Unfinished or not yet written scriptures as he was present in the Christian congregation at the time. Note also that some of those dates are approximate.
Table of the Books of the Bible. Make sure to enlarge that picture so you can see dates where the page got cut off.
The Corinthians would not have had the same bible we have because it was not yet finished. The account of Matthew was written and had enough time to get to them. The Corinthians would have had the scrolls of the bible or manuscripts up to that date.
Because the other scriptures hadn't been written its pretty safe to assume Paul was teaching them from the same scriptures. The account of Jesus's death and Resurrection where in Matthew, so that is probably what he is talking about with Jesus dying for our sins and rising up.