Greek astronomers knew the earth was spherical around the 6th century BC (easily deduced by observing ships sailing in the Mediterranean, as well as the visibility of the stars at different latitudes) and had already demonstrated that the moon was a sphere 150 years before Ceasar's reign. Pliny records that Gaius Sulpicius Gallus enjoyed promotion for winning a battle by correctly predicting an eclipse and timing his military action around it, also before Ceasar's birth. Around the same time, Seleucus (who lived in what we now call Iraq) postulated that the earth orbited the sun rather than vice versa.

Ceasar himself was interested in astronomy, to the extent that he replaced the previous Roman calendar with the Julian Calendar (a modification of the Egyptian Solar calendar), the first to feature a year of 365 days and and include a quadrennial leap year, which endured for 1500 years before the Gregorian Calendar tweaked the leap year scheme for centenary years; considering that the inaccuracy of the Julian calender only amounted to 3 days over 400 years, it error factor was < 0.001%.

Ceasar may not have thought of the earth as just another planet orbiting just another sun in just another galaxy, but it's highly unlikely that he believed the earth to be flat.