You're confusing two different systems of logic. Sentential or propositional logic uses variables to represent whole sentences (hence the name), and connects them with operators such as if-then. Truth tables are used in sentential logic. (First-order) predicate logic extends sentential logic by adding object-predicate constructions (like "x is a cube") and quantifiers ("there is an x such that").
Truth tables can't be used in predicate logic. To see why, consider the sentence
(x) x=x (for all x, x is identical with x). This sentence is logically true. But if you tried to represent it in sentential logic, you would get simply
p is not logically true — one row of its truth table is false.
(Also, logical truth, logical necessity, and tautology are all synonymous. So in sentential logic, finding a row of the truth table where the sentence is false shows that it's not logically true.)
In more-or-less natural English, your sentence is equivalent to the following: "either something is not a cube or everything is a cube." It will probably be easy to see that this is logically true. (Maura ALLEGRANZA's answer gives more detail here.) But that's not yet a formal proof.
Here's a formal proof using reductio ad absurdum:
WTS: (Ex)[Cx -> (y) Cy]
1. Assume for RAA: -(Ex)[Cx -> (y) Cy]
2. (x) -[Cx -> (y) Cy]
3. -[Ca -> (y) Cy]
4. Ca & -(y) Cy
6. -(y) Cy
7. (Ey) -Cy
9. -[Cb -> (y) Cy] (from 2)
10. Cb & -(y) Cy
-><- (8, 11)