Well...Observatories of the kind we know, didn't exist until well after the invention of the telescope between 1600 and 1610. In 1641, Hevelius built an observatory on the roof of his house with a small-aperture, long focus refractor. The National Observatory of France was built in Paris between 1667 - 1671. The Royal Greenwich Observatory was built in 1675.
In the 1500's, Tycho Brahe's 'observatory' at Uraniborg on the Isle of Hveen was built in 1576 and consisted of a large room filled with sextants and other kinds of instruments for making precise angular measurements; but of course no telescopes. Johann Muller ( Regiomontanus) built a smaller, less elaborate observatory in 1471 AD. The last, great Islamic observatory was built by Ulugh Beigh in Samarkand in 1420 AD and was noted for its huge, 40-meter meridian circle.