The 1841 British edition replaced his with this, undoubtedly a typo rather than the result of editing. The subsequent LDS edition (1849) restored the correct “in his day”. The 1841 typesetter was prone to mix up his and this; we have three other examples of this kind of typo in that edition (all three of the others incorrectly replace this with his):
The phrase “in his day” is definitely correct here in Helaman 8:22. For further discussion of this phrase, especially the use of the singular day instead of the plural days, see nearby under Helaman 8:16. Interestingly, the original text has no instances of the phrase “in this day”.
Summary: Maintain the determiner his in Helaman 8:22: “and have rejoiced in his day which is to come”.