End of supported life for an operating system does not mean the operating system and the installed software will stop working. It means they will no longer receive any updates (bugfixes, new features).
This is going to happen to Mint 15 Olivia at the end of January 2014.
Upgrading from Mint 15 to Mint 16 can be performed in two different ways.
Please, read Clem's tutorial
on how they work: How to upgrade to a newer release
. Clem explains both upgrade paths, their pros and their cons, and it is no scret that he is in favour of a clean installation as opposed to an in-place upgrade.
Therefore, you may like to read the tutorial Upgrade linux mint in-place using apt
as well and decide for yourself which upgrade path will be your path.