Does Christ explicitly say that this was his first time on Earth? Is it possible that other historical figures like Buddha and Krishna could be prior incarnations of Jesus? Note I'm specifically asking if Christ said it was exclusive, not other Biblical texts.
No, Christ never said it was His first time on earth.  As a matter of fact, many denominations believe that he appeared several times in Old Testament Scripture, often referred to as the Angel of the Lord.

As far as whether Buddha or Krishna could be prior incarnations, it would go against everything Jesus taught. The teachings of these other religions are very different from Christianity.  Christianity is quite unique in several ways, and certainly isn't compatible with Buddhism (which emphasizes man's journey to enlightenment) or Hinduism (a pantheistic religion).  The idea that these figures could be "other appearances of Christ" would need to completely ignore huge differences in philosophy, and teachings on the nature of man, and man's relationship to the God of Abraham, Isaac, and Jacob (a completely different God than the gods of non Abrahamic religions).