Generating a private key on an air-gapped computer is the way to go. For cold storage your private key should never be on an internet connected device. Best to store it in a durable fashion, like engraving it on a piece of metal.

The deposit address of course you can store all over the place and send Bitcoin to that address for years. You can always check the balance in any online blockchain explorer.

When you're ready to spend it, just enter the private key into your wallet of choice.