Do prime numbers satisfy this? Is this true that $n\log\left(\frac{p_n}{p_{n+1}}\right)$ is bounded, where $p_n$ is the $n$-th prime number?

Seems unbounded:
Let  $g_n = p_{n+1} - p_n$ be the prime gap, then Westzynthius's result (see link below) states that $\lim\sup \left[ g_n/(\log p_n) \right] = \infty$, hence
$$\lim \sup n \log(p_{n+1}/p_n) = \lim \sup n \log (1 + g_n/p_n) = \lim \sup n g_n/ p
 _n =  \lim \sup  g_n/\log n = \infty$$
http://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture
