There is a cut-away diagram of the observatory about 2/3 of the way down this page: https://mindofahitchhiker.com/ulugh-beg-observatory-samarkan.... You can see that the main feature was the quadrant hallway: a narrow vertical open space with a window set at a top corner and quarter-circle measurement track rising from the below-ground levels up the back wall of the building, as in this cut-away ascii art:

      /window
      xxxxxxxxxxxx
    x           |x
    x           |x
    x           /x
    x          /xx
    x         /x x
    x       - xxxx 
    x    --xx xx x
    x---xxxxxxxxxx

then an observer on bottom can measure the angle of elevation to a star visible in the window. As the planet rotates a different slice of the sky can be observed over the course of the night, allowing careful measurements of a large portion of the visible sky.