Captain Rumsey's Lager

Actually it doesn't. Does every Chef need to be a fisherman too? It doesn't work the other way around.

Here's the thing.

The recipe for Captain Rumsey's Lager can be obtained from the Bag of Fishing Treasures which is a reward for completing the fishing daily Crocolisks in the City, which requires a fishing skill of 1.

To answer your question, yes. If you want this achievement you will have to be a fisherman, albeit a very, very poor one.

Edit: Broken link.

Both Ingredients for this one are sold by Kylene in Shattrath.
The recipe is a random drop from the Fishing daily quests, and has a fairly high drop rate.

The drop rate is reported so low because once you learn it you never see it drop again, thus altering the perceived drop chance.