I have never come across a statement linking linearity and monotony - but it seems that for each linear function (positive, negative or even constant slope), the function is monotonic:
I.e. for y $\geq$ x it follows that f(y) $\geq$ f(x).
Is this correct?
Not necessarily monotonic increasing, consider $f:\mathbb{R}\rightarrow\mathbb{R}$ with $f(x)=-x$
clearly $f$ is linear but $1\geq 0$ and $f(1)=-1<0=f(0)$.
In general if $f$ is linear then so is $-f$ and it is impossible that both are monotonic increasing.
It is true however that every linear function is monotonic because every linear function from $\mathbb{R}\rightarrow\mathbb{R}$ takes the form $f(x)=ax$ where $a=f(1)$.