No, the ratio $u'''/u'$ need not be monotone. Since there is no restriction on the fourth derivative, nothing prevents us from adding a spike to $u'''$ that will destroy its monotonicity without violating any other constraints.
More concretely, let $u$ be a function as in the statement. Let $v(x)=u(x)+ \delta \epsilon^3 \sin (2\pi \epsilon^{-1}x)\chi_{[1\le x\le 1+\epsilon]} \tag1$ Ignoring insufficient smoothness* at $1$ and $1+\epsilon$, we see that $v$, $v'$ and $v''$ are within $\delta\epsilon $ of the corresponding $u$ derivatives. Also, $v'''-u''' $ is of order $\delta$. Fix $\delta$ small enough so that $v'''$ is positive. Now make $\epsilon$ very small. The added term in (1) has huge 4th derivative (first positive, then negative), and this makes $v'''$ non-monotone.
(*) Instead of one period of sine function, one should use a smoother compactly supported function, such as convolution of one period of sine with a smooth bump function.