Given sunrise, noon, sunset, longitude and latitude can I calculate the "hight" or "ascension" of the sun?
To the precision that you need, you can calculate the azimuth A and altitude h of the Sun from the following equations:
$$
\tan(A) = \frac{\sin(H)}{\cos(H)\sin(\phi)-\tan(\delta)\cos(\phi)}\\
\sin(h)=\sin(\phi)\sin(\delta)+\cos(\phi)\cos(\delta)\cos(H)
$$
where


A is the azimuth measured from due south (positive towards the west)
H is the hour angle which is the number of degrees before (negative angle) or after (positive) solar noon. Calculate this as 15*1.00274*(time of day - time of noon).
$\phi$ is the latitude (positive for northern hemisphere)
$\delta$ is the declination of the Sun. This can be calculated from the second equation by assuming the altitude of the Sun h=0 when it is rising or setting, and H=15*1.00274*time from sunrise to solar noon. The assumption here is that the time from solar noon to sunset is the same.
h is the altitude of the Sun above the horizon.


All of the above quantities are in degrees. You may need to convert those to radians to perform the trigonometry calculations.

For a more accurate calculation, you would calculate the position of the Sun (its right ascension $\alpha$ and declination $\delta$) and Greenwich Mean Sidereal Time (GMST) based on the date and time throughout the day. Then using the latitude and longitude, you would calculate H, A, and h for various times of the day.