In
H.L. Montgomery, Extreme values of the Riemann zeta function, Comment. Math. Helvetici 52 (1977) 511-518
it was shown in the Corollary on page 511-512 that $$\arg \zeta (s)=\Omega_{\pm}((\log t)^{1-\sigma}(\log \log t)^{1-\sigma})$$ for $1/2<\sigma<1$ and large $t$ as $t \to \infty$. Here, what does the symbol $\Omega{\pm}$ mean? Also, are there any "O" results (i.e.,$\arg \zeta (s)=O(\cdots)$) for $\arg \zeta (s)$ for $1/2<\sigma<1$ and large $t$ as $t \to \infty$?
As in the comments, $f(x)=\Omega_+(g(x))$ usually means that $\limsup_{x\to\infty} \frac{f(x)}{g(x)}>0$ and $f(x)=\Omega_-(g(x))$ usually means that $\liminf_{x\to\infty}\frac{f(x)}{g(x)}<0$. The symbol $\Omega_\pm$ means that both inequalities above holds.
For your second question, yes we have that, assuming the RH
$\arg \zeta(s)=O(\frac{\log t}{\log\log t})$, for $\Re(s)\geq\frac{1}{2}$, as $t \to\infty$.
Unconditionally, we have for $\frac{1}{2}<\sigma<1$
$\arg\zeta(s)=O(\log t),$ as $t\to\infty$.
For a reference you can check the classic: Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics, 97. From Montgomery and Vaughan.