I wonder whether you are:
The letter 'Y'
I'm there when it's cloudy
I'm there when it's sunny
The letter 'Y' appears at the end of both CLOUDY and SUNNY.
And when I ask questions
When spoken out loud, 'Y' is a homophone of 'WHY?' (a question)
I am right on the money
The symbol for the Japanese yen (¥) resembles a Y with two horizontal bars. Plus, as @JMP points out in comments, 'Y' is literally the letter on the RIGHT-hand side of the word MONEY.
I take both sides
In any conversation
To each party in a conversation the other is 'YOU', which begins with 'Y'.
(As pointed out later by the OP, 'I take both sides' further refers to this letter's ability to be both a vowel and a consonant.)
I am not so common
The letter 'Y' is not one of the most frequently used in the English language.
(Another interpretation pointed out later by the OP concerns the element Yttrium - whose Periodic Table symbol is 'Y' - being one of the "rare-earth elements".)
Yet very very common
Approximately half of the world's population carry a Y-chromosome (males), which makes it very widespread indeed... This line also contains 3 Y's (Yet very very common) as if for emphasis.
As for the title:
A martini is typically served in a Y-shaped glass! (Thanks @JMP)
And to answer in rhyme:
Don't be so shy
I am letter Y!