It's not a common idiom but it would be understood and sounds quite good.

"came on shift" isn't something we would say about the say. We might say, "the sun finally came out after..."

I would use “The sun finally came out after days of rain.” to express "连日阴雨后终于出太阳了。"

No, it sounds bad. It's a weak metaphor comparing the sun to an hourly worker. It will be understood because the sentence "The sun finally came _______ after days of rain" will be understood no matter how you fill the blank.

Hello, the sentence does not sound natural and is not something that we as natives would say. The phrase 'came on shift' is awkward in this context. You could have said something like this: "The sun finally showed its pretty face after days of raining." Hope this helps.