Step 1: Instead of listing candidates for each statement separately, set up the two-digit number algebraically as $10a+b$, where $a$ is the tens digit ($1$ to $9$) and $b$ is the units digit ($0$ to $9$), and test each statement using modular arithmetic.
Step 2: Statement I says $a+b=9$. This equation alone has multiple solutions: $(a,b)$ can be $(1,8), (2,7), (3,6), (4,5), (5,4), (6,3), (7,2), (8,1), (9,0)$, giving nine different two-digit numbers ($18, 27, 36, 45, 54, 63, 72, 81, 90$). The number is not unique, so Statement I alone is NOT sufficient.
Step 3: Statement II says the number is divisible by 9 and $a=4$. Divisibility by 9 means $10a+b\equiv0\pmod9$, and since $10\equiv1\pmod9$, this simplifies to $a+b\equiv0\pmod9$.
Step 4: Substitute $a=4$: $4+b\equiv0\pmod9\Rightarrow b\equiv-4\equiv5\pmod9$. Since $b$ must be a single digit ($0$ to $9$), the only valid solution is $b=5$.
Step 5: This gives the unique two-digit number $10(4)+5=45$. Statement II alone determines the number uniquely, so Statement II alone IS sufficient, while Statement I alone is not.
\[\boxed{\text{Statement II alone is sufficient, but Statement I alone is not sufficient.}}\]
Correct option: (2) Statement II alone is sufficient, but Statement I alone is not sufficient.