No, it is certainly not true that Julius Caesar read Sun Tzu's The Art of War. I am not sure where you would have heard this.

The Romans in Caesar's time were only vaguely aware of the existence of China and they did not have access to any classic texts written in the Chinese language. Moreover, even if Caesar had somehow managed to gain access to a copy of The Art of War, he certainly would not have been able to read it, since no Latin or Greek translation of the work is known to have existed and Caesar certainly could not read or understand Chinese.

Even Pliny the Elder (lived c. 23 – 79 CE), who lived several generations after Caesar at a time when somewhat better information about China was available to the Romans due to the intervening decades of trade between Rome and East Asia and who seems to have been generally better read than Caesar on the subject of distant lands anyway, still has relatively little to say about the Chinese in his Natural History 20.54–55.

Pliny's information about the Chinese basically amounts to knowledge about Chinese silk production, a general impression of the Chinese as "gentle" in temperament and isolationist in policy, and some vague and not-particularly-accurate information about Chinese geography. He shows no awareness of the existence of any Chinese literature, let alone The Art of War specifically.

Literate Romans like Caesar and Pliny who could read Greek had access to much better information about India than they did about China, primarily thanks to Greek writers like Aristoboulos of Kassandreia (lived c. 375 – 301 BCE) and Megasthenes (lived c. 350 – c. 290 BCE), who had actually gone to parts of northwest India and wrote accounts of what the country was like. Nonetheless, even much of what ancient Greek and Roman authors say about India is wildly inaccurate and fabulous.