The partial sums, normalized by $\sqrt{p}$, are unbounded, as one varies $a$ over all invertible classes modulo $p$n and lets $p$ go to infinity.  This follows from results in <a href="http://front.math.ucdavis.edu/1410.7892">this paper</a>, which also has more precise information on the distribution of the partial sums, including real and imaginary parts. 

(By the way, it is more usual to write $ax+x^{-1}$ in the phase of the Kloosterman sum, instead of $x+ax^{-1}$; this emphasizes that it is, as a function of $a$, the discrete Fourier transform of $e_p(x^{-1})$).