By "cold storage" I am going to assume you mean a private key kept offline. If that is the case you should start by installing wallet software. Electrum or Core are good choices. It's likely best to go with the software that created the wallet you are storing.
It will need to sync, so settle in for a few days at least. During that time set up an exchange account linked to your bank. It should be an exchange that has a business license. NEVER send to an unlicensed business.
Now you are set! Send the BTC to the exchange, sell it, withdraw the fiat currency to your bank.
Electrum wallet does not need days to sync with network, for the first synchronisation time, after installing. It just takes a few minutes to completely sync with network, in my experience.
You are right with Bitcoin Core wallet, it takes days to sync, and requires huge storage capacity. I tried to install and sync that wallet, then failed (due to my computer storage capacity). After that, I moved to Electrum, that helps (lighter and faster).