  This paper generalizes the result of Sarnak and Ubis \cite{sarnak-ubis} about
non-concentration of primes in horocycle orbits on $PSL_2(\mathbb{Z})
\backslash PSL_2(\mathbb{R})$ to any lattice in $PSL_2(\mathbb{R})$. The proof
combines the asymptotic result of Str\"ombergsson \parencite{strombergsson} and
Venkatesh's method \parencite{venkatesh} with the approach of Sarnak and Ubis
of approximating horocycle pieces with periodic horocycles. The key step is to
establish a dichotomy between $\{\xi h(t), t \in [0, T] \}$ having good
equidistribution in $\Gamma \backslash PSL_2(\mathbb{R})$ and it being
approximable by closed horocycle pieces with small period. In a follow-up
paper, a similar approach will be used to show equidistribution of $\xi
h(n^{1+\gamma})$ for small $\gamma>0$, generalizing Venkatesh's result
\parencite{venkatesh} to non-compact $\Gamma$.
