It's not entirely true that when you take the reciprocals of both sides, you change "$\lt$" to "$\gt$" and vice-versa. If $a$ and $b$ are both positive or if they are both negative, and $a<b$, then $1/a>1/b$. But if one is positive and the other negative, and $a<b$ (which means $a$ must be the one that's negative) then $1/a<1/b$; the direction doesn't get reversed.
Generally if $a<b$ and $g$ is a strictly decreasing function, then $g(a)>g(b)$. That's actually the definition of the concept of "strictly decreasing function". So the fact that when you multiply by a negative number, you invert the inequality relation, is the same as saying that multiplication by a negative number is a strictly decreasing function. For example, if $g(x) = -5x$ for all values of $x$, then $g$ is a strictly decreasing function. If $g(x)=1/x$, then the restriction of $g$ to the positive numbers is a strictly decreasing function, and the restriction of $g$ to the negative numbers is a strictly decreasing function, but $g$, over its whole domain, is not a strictly decreasing function.
$\arccos$ is a strictly decreasing function: as a number increases from $-1$ to $1$, its arccosine decreases, i.e. if $a<b$, then $\arccos a>\arccos b$.
One thing that will tell you that a function is strictly decreasing is that it's derivative is everywhere negative and its domain has no gaps. The reciprocal function has an everywhere negative derivative, but its domain has a gap at $0$.