How do I determine the phase response of a high pass filter?
The phase response is just the argument of the transfer function (just as the magnitude response is the absolute value). 

The argument of a quotient is the argument of the numerator minus the argument of the denominator, i.e.,

$$
\phi = \operatorname{arg} \frac{j\omega}{j\omega+\frac{1}{RC}} = \operatorname{arg}(j\omega) - \operatorname{arg}(j\omega+\frac{1}{RC}) = \frac{\pi}{2} - \operatorname{atan2}(\omega, 1/RC )
$$
which is essentially just what you wrote.

Note that I did not write \$\operatorname{tan}^{-1}(y/x)\$ but \$\operatorname{atan2(y,x)}\$ because the former is only correct if \$x\$ is positive. It is incorrect when \$x\$ is zero or negative. In your case here, when \$\omega\$ is always positive that makes no difference, but I think it is good practice to use atan2.