Converting Time Shift to Phase Shift
Indeed, you can apply the same technique:

As you said, given a sine signal with time shift $t_1$, which is $x(t)=\sin(w(t-t_1))$, this becomes $x(t)=\sin(w t - wt_1)=\sin(wt+\phi)$ and $\phi=-wt_1$.

Now, if you encounter a signal $x(t)=\sin(\frac{\pi}{4} t - \frac{\pi}{4})$ you identify $w=\frac{\pi}{4}$ and $\phi=-\frac{\pi}{4}$. What remains is to solve $\phi=-w t_1$ for $t_1$ and you have it.