+600 Take the entire thing mod $10$.
It's then just $9(1^{2009}+...+9^{2009})\pmod{10}$.
And $1^{2009}=(-9)^{2009}=-9^{2009}\pmod{10}$.
Similarly, $2^{2009}=-8^{2009}\pmod{10}$, etc.
So everything cancels and it becomes $9\cdot 5^{2009}\pmod{10}$. Note that $5^n$ is always $5$ modulo $10$. Then the answer is $9\cdot 5=5\pmod{10}$.
(mod 10 is just the remainder when divided by 10)
