Given $g,g^t$ in an RSA group modulo $N=pq$, is it hard to compute $g^{t^{-1}}$? Suppose we have an RSA groug $G=\mathbb{Z}^{*}_{N}$, where $N=pq$ , where $p,q$ are primes. Let $g$ be a random element of $G$ and $t\in \mathbb{Z}^{*}_{N}$. Having $g$ and $g^t$, it seems to be very hard to find $g^{t^{-1}}$, supposing discrete logarithm problem (DLP) is hard. My question is that is this really hard? 

In a prime order group, let's say in $\mathbb{Z}^{*}_p$ , $g^{t^{-1}}$ = $g^{p-2}$ : here we don`t have to face DLP, so finding $g^{t^{-1}}$ is easy.
I know that I am not exactly answering your question, but I am pointing you in a potentially interesting research direction. Your question is not standard in the area of discrete log and Diffie-Hellman problems since you are considering a cyclic group of order $p\cdot q$ where $p$ and $q$ are primes. (Typically, we like looking at group of prime order.)

Your question is actually asking whether you can reduce this problem to solving the discrete log problem. I don't have any answer to that, but it reminds of an old paper by Biham, Boneh and Reingold that shows that the Diffie-Hellman problem modulo $N=pq$ (for Blum integers) is actually equivalent to factoring. This isn't the same thing, but there's something similar. The paper is here: https://omereingold.files.wordpress.com/2014/10/cgdh.pdf.