Is it correct to say that an algorithm ALG is an O(1)-approximation algorithm?
When we say that ALG is an $O(1)$-approximation algorithm, we meant that there exists a constant $C$ such that ALG is a $C$-approximation algorithm. Sometimes we don't care about the exact value of $C$, which is why we use the $O(1)$ notation.

You don't get to choose the constant – rather you are only given that such a constant exists.