You won’t find Podman in the default repositories configured on Linux Mint. To install it using apt you will need to configure a third party repository that provides a Debian package for it. The Kubic project provides recent packages for Podman (in addition to other useful tools for working with containers). It’s worth noting that the project does not provide a specific Linux Mint build but, being a Debian-based distribution, the Ubuntu build seems to work just fine.

I use the Ubuntu 20.04 source in this example. You could also use the other Debian-based build targets listed though your mileage may vary.

SOURCE="xUbuntu_20.04"
# Adds the Kubic Debian repository to the systems apt sources.
echo "deb https://download.opensuse.org/repositories/devel:/kubic:/libcontainers:/stable/${SOURCE}/ /" | sudo tee /etc/apt/sources.list.d/devel:kubic:libcontainers:stable.list
# Adds the repository encryption key for verifying package signatures.
curl -L "https://download.opensuse.org/repositories/devel:/kubic:/libcontainers:/stable/${SOURCE}/Release.key" | sudo apt-key add -

sudo apt update
sudo apt install podman

Now you can verify your installation.

podman run hello-world