Yes, such a closed geodesic always exists. See Theorem 1.1 of this paper by Basmajian, Parlier, and Souto (which I found by searching under the term "density of closed geodesics on a hyperbolic surface").

Now, to be honest, what you want is simpler than what is proved in that paper. Namely, it is well known that the geodesic flow has a dense (non-closed) geodesic $\gamma : \mathbb{R} \to S$. Take a subsegment $\gamma | [-M,+M]$, close it off with a uniformly short segment, and straighten to get a closed geodesic $\gamma_M$. For each $\epsilon>0$, if $M$ is sufficiently large then $\gamma_M$ is $\epsilon$-dense.

Yes, such a closed geodesic always exists. See Theorem 1.1 of this paper by Basmajian, Parlier, and Souto (which I found by searching under the term "density of closed geodesics on a hyperbolic surface").

Now, to be honest, what you want is simpler than what is proved in that paper. Namely, it is well known that the geodesic flow has a dense (non-closed) geodesic $\gamma : \mathbb{R} \to X$. Take a subsegment $\gamma | [-M,+M]$, close it off with a uniformly short segment --- namely one of length at most the diameter of $X$ --- and straighten. If $M$ is sufficiently large then the result of straightening is a closed geodesic $\gamma_M$. Furthermore, for each $\epsilon>0$, if $M$ is sufficiently large then $\gamma_M$ is $\epsilon$-dense meaning that it hits every $\epsilon$-ball in $X$.