Your programs will continue to work but no more updates or security fixes will be made available for them after April. Users of Linux Mint 13 are therefor recommended to migrate to Linux Mint 18.1 (which will be supported until April 2021) or Linux Mint 17.3 (supported until April 2019) before May. For all available supported releases and an overview of support periods see https://linuxmint.com/download_all.php.
Upgrade instructions: To upgrade to a newer release of Linux Mint, backup your important personal files and do a fresh installation. You can not upgrade in place (without reinstalling) as you can do with newer releases of Linux Mint. You may refer to the How to upgrade to a newer release tutorial or ask for support from others on the Installation & Boot forum. If you have a dual boot installation (Linux Mint alongside another operating system) I recommend you ask for support on the forum if you are unsure of the steps.
If you have an older computer that can only run 32-bit software and it doesn't officially support PAE—like a computer with an Intel Pentium M processor—the chance is good you can use Linux Mint 17.x and 18.x anyway. Instructions for this from the release notes: "To boot Linux Mint on CPU which do not officially support PAE please use the 'Start Linux Mint with PAE forced' option from the boot menu." If it doesn't work on your computer you can instead use LMDE 2 as that works with any 32-bit processor.
Linux Mint 17.x and 18.x do require more free disk space for the installation (9 GB instead of 5 GB for Linux Mint 13) but otherwise the system requirements are comparable. The MATE edition and certainly the Xfce edition would be an option for older computers.
If MATE or Xfce are too heavy you can also switch to LXDE. Just install the packages
lxde-common. I recommend you do that with the following command to also get the additionally recommended LXDE packages installed:
apt install --install-recommends lxde lxde-common
Then logout and on the login screen select to login to a LXDE session (use the gearwheel icon in the top right of the login box).