I do consider this incorrect, because 'a couple of cups of tea' can also mean more than 2 cups. Grammatically it is not incorrect, but it does not really make sense to use this sentence. Definitely not if you want people to know that you mean 2 cups. You can indeed say (Like Evelin said) "I drank a couple of cups of tea and went to work". But that still doesn't sound natural to me. "I drank some tea and went to work" is what I would say. I hope this helps

It's not incorrect, but it sounds more informal. If I'm ordering at a café, I'd only use "two cups of tea". But I could say something like: "This morning I prepared a couple of cups of tea and went to work".