Step 1: Analyzing the order of solubility.
The solubility of alkali metal carbonates in water increases as the size of the cation increases. Larger cations like Cs$^+$ and K$^+$ form weaker ionic bonds with the carbonate ion, making the compounds more soluble in water.
Step 2: Conclusion.
Thus, the correct order for solubility in water is Cs$_2$CO$_3$>K$_2$CO$_3$>Na$_2$CO$_3$>Li$_2$CO$_3$, corresponding to option (B).