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In Hakuouki, he's 22, (according to the modern Western way of calculating age.) I'll quote ochimizu's post on Hakuouki and history explaining why:
And other thing; the reason why Okita's birthday in Hakuouki is listed as 1842 and not 1844 — most sources cite that he was born in 1844, the same year as Saitou and Heisuke. Hakuouki portrays him as being older than them both, using 1842 as his birthyear. They also list his birthday as "Summer," with no date given. The reason for this is that according to Okita family legend, he was born during either a terrible thunderstorm or during a solar eclipse. In June of 1842, there was a solar eclipse.
Another little detail, the characters in Hakuouki in 1864 would regard him as 23. In traditional East-Asian age reckoning, a baby is aged 1 on its date of birth, and gains an extra year in age every New Year's Day. (And yes, a baby born on the last day of the Old Year turns 2 on the second day of its life.)
