If $f(n)=O(g(n))$ can I say there exists a $n_0$ for a specific $c$?
Let's take as an example $f(n) = g(n) = n$ and $c = 1/2$. Does there exist an $n_0$ such that $f(n) \leq cg(n)$ for all $n \geq n_0$?
In your case, you know more: $\log n = o(n)$. This means that
$$
\lim_{n\to\infty} \frac{\log n}{n} = 0,
$$
which implies that for every $c$ there exists $n_0$ such that $\frac{\log n}{n} \leq c$ for all $n \geq n_0$.