How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?

Integrate by parts:

\begin{align} \int_x^{x+1}\sin(e^t)dt & =\int_x^{x+1}e^{-t}d(-\cos(e^t)) \\ & =e^{-x}\cos e^x-e^{-x-1}\cos e^{x+1}-\int_x^{x+1}e^{-t}\cos e^{t}dt\\ & =e^{-x}\cos e^x-e^{-x-1}\cos e^{x+1}-\int_x^{x+1}e^{-2t}d\sin e^{t}\\ & =e^{-x}\cos e^x-e^{-x-1}\cos e^{x+1}-e^{-2(x+1)}\sin e^{x+1}\\ & \hphantom{={}}+e^{-2x}\sin e^x+2\int_x^{x+1}e^{-2t}\sin e^tdt.\end{align}

From here we see that $e^x \int_x^{x+1}\sin(e^t)dt$ is bounded by $1+1/e+O(e^{-x})$ and $1+1/e\approx 1.368$ can not be improved, since both $\cos e^x$ and $-\cos e^{x+1}$ may be almost equal to 1: if $e^x=2\pi n$ for large integer $n$, then $e^{x+1}=2\pi e n$, we want this to be close to $\pi+2\pi k$, i.e., we want $en$ to be close to $\frac12+k$.

This is possible since $e$ is irrational. Moreover, $e$ is so special number that you may find explicit $n$ for which $en$ is nearly half-integer: $n=m!/2$ for large even $m$ works. Indeed, $e=\sum_{i=0}^{m-1}1/i!+1/m!+o(1/m!)$ yields $em!/2=\text{integer}+1/2+\text{small}$.

Here is a method that will allow one to find the exact upper and lower bounds on $g(z)$ over $z>0$ with any degree of accuracy.

Take any real $z>0$. Since \begin{equation*} \frac1y=\int_0^\infty dt\,e^{-y t} \end{equation*} for any real $y>0$, we have \begin{align*} \frac{g(z)}z &=\int_z^{e z} dy\, \frac{\sin y}y \\ &=\int_0^\infty dt\,\int_z^{e z} dy\,e^{-y t}\sin y \\ &=\int_0^\infty dt\, \Big( \frac{e^{-t z} (\cos z+t \sin z)}{t^2+1} -\frac{e^{-e t z} (\cos ez+t \sin ez)}{t^2+1}\Big) \\ &=I_1(z) \cos z+I_2(z)\sin z -I_1(ez) \cos ez-I_2(ez)\sin ez, \tag{1} \end{align*} where \begin{align*} I_1(z)&:=\int_0^\infty dt\,\frac{e^{-t z}}{t^2+1}, \\ I_2(z)&:=\int_0^\infty dt\,\frac{e^{-t z}t}{t^2+1}. \end{align*} Next, letting $c_1$ and $c_2$ denote functions with values in $(0,1)$, we have \begin{align*} I_1(z)&=\frac1z\,\int_0^\infty du\,\frac{e^{-u}}{1+u^2/z^2} \\ &=\frac1z\,\int_0^\infty du\,e^{-u} -\frac1z\,\int_0^\infty du\,\frac{u^2e^{-u}}{z^2+u^2} \\ &=\frac1z-\frac{2c_1(z)}{z^3}; \end{align*} at the last step here, we used the inequality $z^2+u^2>z^2$ for $u>0$;
similarly, \begin{align*} I_2(z)&=\frac1{z^2}-\frac{3c_2(z)}{z^4}. \end{align*} So, by (1), \begin{equation*} g=h+r, \end{equation*} where \begin{equation*} h(z):=\cos z-\tfrac1e\,\cos ez \end{equation*} and \begin{equation*} r(z):=-\frac{2c_1(z)}{z^2}\, \cos z-\frac{3c_2(z)}{z^3}\,\sin z +\frac{2c_1(ez)}{e^3z^2}\, \cos ez+\frac{3c_2(2z)}{e^4z^3}\,\sin ez \end{equation*} is the "remainder", so that \begin{equation*} |r(z)|<\frac{2.1}{z^2}+\frac{3.1}{z^3}, \end{equation*} which can be made however small if $z$ is large enough.

On the other hand, since $e$ is irrational, we will have \begin{equation*} \sup_{z>0}h(z)=-\inf_{z>0}h(z)=1+1/e=1.367\dots \end{equation*} (which is somewhat close to your value $1.4$).

So, to compute $\sup_{z>0}g(z)$ and $\inf_{z>0}g(z)$ with any degree of accuracy, it suffices to be able to compute $\sup_{z\in(0,a]}g(z)$ and $\inf_{z\in(0,a]}g(z)$ with any degree of accuracy for any given real $a>0$, which can be done by (say) the interval arithmetic method, using the formula $g(z)=z(\text{Si}(e z)-\text{Si}(z))$ and the monotonicity of the function $\text{Si}$ on each of the intervals of the form $[k\pi,(k+1)\pi]$ for $k=0,1,\dots$.

This can be done with help of Maple in such a way. First, we find the estimated expression explicitly by

 a := (exp(x)*int(sin(exp(t)), t = x .. x + 1) assuming x::real;

$ {{\rm e}^{x}} \left( -{\it Si} \left( {{\rm e}^{x}} \right) +{\it Si} \left( {{\rm e}^{x+1}} \right) \right) $

In fact, the integral is reduced to another integrals. Next, the asymptotics of $a$ is found. Maple is not able to find this asymptotics directly so the change $x=\log y$ should be used:

asympt(simplify(eval(a, x = log(y))), y, 2);

$-{\frac {\cos \left( y{\rm e} \right) }{{\rm e}}}+\cos \left( y \right) +O \left( {y}^{-1} \right) $

Now we return to $x$ by

eval(%, y = exp(x));

$-{\frac {\cos \left( {{\rm e}^{x}}{\rm e} \right) }{{\rm e}}}+\cos \left( {{\rm e}^{x}} \right) +O \left( \left( {{\rm e}^{x}} \right) ^{-1} \right) $

The rest is as in the Fedor Petrov's answer.