Proper cruise etiquette dictates that when at the pool, buy your drinks at the bar. I do bring some onboard with me but for cabin consumption. Carrying one of those big mugs from the gas station out to the pool is in poor taste. If you do bring a beer from the cabin to the pool, it should be of the same variety that is sold at the bar. You don't come out the the pool carying a 16 OZ Natural Ice!
MSBJ, my cooler has a strap and is narrow 6 inches, I put my camera stuff, docs, wallet, birth certs, ect... to carry on board. Beers and booze go in my luggage.
I bought this bottle of Vodka in the shop onboard when they used to let you take it to your cabin.