Well, it looks like it should sound like "cajita", as in "cajita feliz" ("happy meal"), but actually his name is pronounced [kadʒita], not [kaxita]; think "callita", like a small digging stick, but pronounced with a Rioplatense accent. Japanese transliteration is misleading in Spanish!
(You could kind of make the same complaint about pronouncing "tomato" as [tə'meɪɾoʊ], since the second vowel in the Nahuatl word is very clearly [a], but I think you can make a reasonable case that "tomato" is an English word with several centuries of history whose pronunciation has simply changed, while Kajita is not a Spanish word, but rather the name of a Japanese guy who's in fact not even dead yet. So if you pronounce it [kaxita], you're just mispronouncing it, that's all.)
(You could kind of make the same complaint about pronouncing "tomato" as [tə'meɪɾoʊ], since the second vowel in the Nahuatl word is very clearly [a], but I think you can make a reasonable case that "tomato" is an English word with several centuries of history whose pronunciation has simply changed, while Kajita is not a Spanish word, but rather the name of a Japanese guy who's in fact not even dead yet. So if you pronounce it [kaxita], you're just mispronouncing it, that's all.)