I'm sure that you intended to write "fed up with", meaning "very tired of and annoyed with".
The speaker wants some rain.
Could do with = could use; it's a common idiomatic expression. It sometimes is to be taken literally. If a farmer with very dry fields says "we could do with some rain," he means that his crops are in dire need of it. If I say, "I could do with some ice cream," that's obviously not something I need, just something I think I'd like to have right now.