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It is possible that when you created your Droplet, you had an SSH key saved into your DO account (https://cloud.digitalocean.com/account/security).
These SSH keys are usually device specific, meaning that while the public key (~/.ssh/id_rsa.pub) is stored on your DO account, the private key is only on your computer at home. I assume the ‘lab’ computer is a class (shared) computer, which means you could put your private key on it (~/.ssh/id_rsa). Although you should probably give some thought to how secure the shared computer is first, as that key could provide direct access to your VPS.