Can an intersection of two context-free languages be an undecidable language?
Context-free languages are decidable, and decidable languages are
closed under intersection.

So, though the intersection of two CF languages may not be CF, it is decidable.

Remarks on your example:


$\emptyset=\{\}\neq$ $\{0\}$
$L_1\cap\emptyset=\emptyset$  which is context-free.
You cannot prove your claim, because it is wrong
the empty language is decidable: the answer is always "no, this string is not in the empty set".